3956 lines
126 KiB
Fortran
3956 lines
126 KiB
Fortran
c
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c L-BFGS-B is released under the “New BSD License” (aka “Modified BSD License”
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c or “3-clause license”)
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c Please read attached file License.txt
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c
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c=========== L-BFGS-B (version 3.0. April 25, 2011 ===================
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c
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c This is a modified version of L-BFGS-B. Minor changes in the updated
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c code appear preceded by a line comment as follows
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c
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c c-jlm-jn
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c
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c Major changes are described in the accompanying paper:
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c
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c Jorge Nocedal and Jose Luis Morales, Remark on "Algorithm 778:
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c L-BFGS-B: Fortran Subroutines for Large-Scale Bound Constrained
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c Optimization" (2011). To appear in ACM Transactions on
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c Mathematical Software,
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c
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c The paper describes an improvement and a correction to Algorithm 778.
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c It is shown that the performance of the algorithm can be improved
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c significantly by making a relatively simple modication to the subspace
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c minimization phase. The correction concerns an error caused by the use
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c of routine dpmeps to estimate machine precision.
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c
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c The total work space **wa** required by the new version is
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c
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c 2*m*n + 11m*m + 5*n + 8*m
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c
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c the old version required
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c
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c 2*m*n + 12m*m + 4*n + 12*m
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c
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c
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c J. Nocedal Department of Electrical Engineering and
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c Computer Science.
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c Northwestern University. Evanston, IL. USA
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c
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c
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c J.L Morales Departamento de Matematicas,
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c Instituto Tecnologico Autonomo de Mexico
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c Mexico D.F. Mexico.
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c
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c March 2011
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c
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c=============================================================================
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subroutine setulb(n, m, x, l, u, nbd, f, g, factr, pgtol, wa, iwa,
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+ task, iprint, csave, lsave, isave, dsave)
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character*60 task, csave
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logical lsave(4)
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integer n, m, iprint,
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+ nbd(n), iwa(3*n), isave(44)
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double precision f, factr, pgtol, x(n), l(n), u(n), g(n),
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c
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c-jlm-jn
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+ wa(2*m*n + 5*n + 11*m*m + 8*m), dsave(29)
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c ************
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c
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c Subroutine setulb
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c
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c This subroutine partitions the working arrays wa and iwa, and
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c then uses the limited memory BFGS method to solve the bound
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c constrained optimization problem by calling mainlb.
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c (The direct method will be used in the subspace minimization.)
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c
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c n is an integer variable.
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c On entry n is the dimension of the problem.
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c On exit n is unchanged.
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c
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c m is an integer variable.
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c On entry m is the maximum number of variable metric corrections
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c used to define the limited memory matrix.
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c On exit m is unchanged.
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c
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c x is a double precision array of dimension n.
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c On entry x is an approximation to the solution.
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c On exit x is the current approximation.
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c
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c l is a double precision array of dimension n.
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c On entry l is the lower bound on x.
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c On exit l is unchanged.
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c
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c u is a double precision array of dimension n.
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c On entry u is the upper bound on x.
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c On exit u is unchanged.
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c
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c nbd is an integer array of dimension n.
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c On entry nbd represents the type of bounds imposed on the
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c variables, and must be specified as follows:
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c nbd(i)=0 if x(i) is unbounded,
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c 1 if x(i) has only a lower bound,
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c 2 if x(i) has both lower and upper bounds, and
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c 3 if x(i) has only an upper bound.
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c On exit nbd is unchanged.
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c
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c f is a double precision variable.
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c On first entry f is unspecified.
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c On final exit f is the value of the function at x.
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c
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c g is a double precision array of dimension n.
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c On first entry g is unspecified.
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c On final exit g is the value of the gradient at x.
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c
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c factr is a double precision variable.
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c On entry factr >= 0 is specified by the user. The iteration
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c will stop when
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c
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c (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch
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c
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c where epsmch is the machine precision, which is automatically
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c generated by the code. Typical values for factr: 1.d+12 for
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c low accuracy; 1.d+7 for moderate accuracy; 1.d+1 for extremely
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c high accuracy.
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c On exit factr is unchanged.
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c
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c pgtol is a double precision variable.
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c On entry pgtol >= 0 is specified by the user. The iteration
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c will stop when
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c
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c max{|proj g_i | i = 1, ..., n} <= pgtol
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c
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c where pg_i is the ith component of the projected gradient.
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c On exit pgtol is unchanged.
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c
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c wa is a double precision working array of length
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c (2mmax + 5)nmax + 12mmax^2 + 12mmax.
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c
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c iwa is an integer working array of length 3nmax.
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c
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c task is a working string of characters of length 60 indicating
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c the current job when entering and quitting this subroutine.
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c
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c iprint is an integer variable that must be set by the user.
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c It controls the frequency and type of output generated:
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c iprint<0 no output is generated;
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c iprint=0 print only one line at the last iteration;
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c 0<iprint<99 print also f and |proj g| every iprint iterations;
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c iprint=99 print details of every iteration except n-vectors;
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c iprint=100 print also the changes of active set and final x;
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c iprint>100 print details of every iteration including x and g;
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c When iprint > 0, the file iterate.dat will be created to
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c summarize the iteration.
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c
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c csave is a working string of characters of length 60.
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c
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c lsave is a logical working array of dimension 4.
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c On exit with 'task' = NEW_X, the following information is
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c available:
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c If lsave(1) = .true. then the initial X has been replaced by
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c its projection in the feasible set;
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c If lsave(2) = .true. then the problem is constrained;
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c If lsave(3) = .true. then each variable has upper and lower
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c bounds;
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c
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c isave is an integer working array of dimension 44.
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c On exit with 'task' = NEW_X, the following information is
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c available:
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c isave(22) = the total number of intervals explored in the
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c search of Cauchy points;
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c isave(26) = the total number of skipped BFGS updates before
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c the current iteration;
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c isave(30) = the number of current iteration;
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c isave(31) = the total number of BFGS updates prior the current
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c iteration;
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c isave(33) = the number of intervals explored in the search of
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c Cauchy point in the current iteration;
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c isave(34) = the total number of function and gradient
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c evaluations;
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c isave(36) = the number of function value or gradient
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c evaluations in the current iteration;
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c if isave(37) = 0 then the subspace argmin is within the box;
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c if isave(37) = 1 then the subspace argmin is beyond the box;
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c isave(38) = the number of free variables in the current
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c iteration;
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c isave(39) = the number of active constraints in the current
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c iteration;
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c n + 1 - isave(40) = the number of variables leaving the set of
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c active constraints in the current iteration;
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c isave(41) = the number of variables entering the set of active
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c constraints in the current iteration.
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c
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c dsave is a double precision working array of dimension 29.
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c On exit with 'task' = NEW_X, the following information is
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c available:
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c dsave(1) = current 'theta' in the BFGS matrix;
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c dsave(2) = f(x) in the previous iteration;
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c dsave(3) = factr*epsmch;
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c dsave(4) = 2-norm of the line search direction vector;
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c dsave(5) = the machine precision epsmch generated by the code;
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c dsave(7) = the accumulated time spent on searching for
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c Cauchy points;
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c dsave(8) = the accumulated time spent on
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c subspace minimization;
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c dsave(9) = the accumulated time spent on line search;
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c dsave(11) = the slope of the line search function at
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c the current point of line search;
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c dsave(12) = the maximum relative step length imposed in
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c line search;
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c dsave(13) = the infinity norm of the projected gradient;
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c dsave(14) = the relative step length in the line search;
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c dsave(15) = the slope of the line search function at
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c the starting point of the line search;
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c dsave(16) = the square of the 2-norm of the line search
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c direction vector.
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c
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c Subprograms called:
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c
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c L-BFGS-B Library ... mainlb.
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c
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c
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c References:
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c
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c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
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c memory algorithm for bound constrained optimization'',
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c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.
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c
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c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: a
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c limited memory FORTRAN code for solving bound constrained
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c optimization problems'', Tech. Report, NAM-11, EECS Department,
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c Northwestern University, 1994.
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c
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c (Postscript files of these papers are available via anonymous
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c ftp to eecs.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)
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c
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c * * *
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c
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c NEOS, November 1994. (Latest revision June 1996.)
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c Optimization Technology Center.
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c Argonne National Laboratory and Northwestern University.
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c Written by
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c Ciyou Zhu
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c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
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c
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c
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c ************
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c-jlm-jn
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integer lws,lr,lz,lt,ld,lxp,lwa,
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+ lwy,lsy,lss,lwt,lwn,lsnd
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if (task .eq. 'START') then
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isave(1) = m*n
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isave(2) = m**2
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isave(3) = 4*m**2
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isave(4) = 1 ! ws m*n
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isave(5) = isave(4) + isave(1) ! wy m*n
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isave(6) = isave(5) + isave(1) ! wsy m**2
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isave(7) = isave(6) + isave(2) ! wss m**2
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isave(8) = isave(7) + isave(2) ! wt m**2
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isave(9) = isave(8) + isave(2) ! wn 4*m**2
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isave(10) = isave(9) + isave(3) ! wsnd 4*m**2
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isave(11) = isave(10) + isave(3) ! wz n
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isave(12) = isave(11) + n ! wr n
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isave(13) = isave(12) + n ! wd n
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isave(14) = isave(13) + n ! wt n
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isave(15) = isave(14) + n ! wxp n
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isave(16) = isave(15) + n ! wa 8*m
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endif
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lws = isave(4)
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lwy = isave(5)
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lsy = isave(6)
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lss = isave(7)
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lwt = isave(8)
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lwn = isave(9)
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lsnd = isave(10)
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lz = isave(11)
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lr = isave(12)
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ld = isave(13)
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lt = isave(14)
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lxp = isave(15)
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lwa = isave(16)
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call mainlb(n,m,x,l,u,nbd,f,g,factr,pgtol,
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+ wa(lws),wa(lwy),wa(lsy),wa(lss), wa(lwt),
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+ wa(lwn),wa(lsnd),wa(lz),wa(lr),wa(ld),wa(lt),wa(lxp),
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+ wa(lwa),
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+ iwa(1),iwa(n+1),iwa(2*n+1),task,iprint,
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+ csave,lsave,isave(22),dsave)
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return
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end
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c======================= The end of setulb =============================
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subroutine mainlb(n, m, x, l, u, nbd, f, g, factr, pgtol, ws, wy,
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+ sy, ss, wt, wn, snd, z, r, d, t, xp, wa,
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+ index, iwhere, indx2, task,
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+ iprint, csave, lsave, isave, dsave)
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implicit none
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character*60 task, csave
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logical lsave(4)
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integer n, m, iprint, nbd(n), index(n),
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+ iwhere(n), indx2(n), isave(23)
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double precision f, factr, pgtol,
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+ x(n), l(n), u(n), g(n), z(n), r(n), d(n), t(n),
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c-jlm-jn
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+ xp(n),
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+ wa(8*m),
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+ ws(n, m), wy(n, m), sy(m, m), ss(m, m),
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+ wt(m, m), wn(2*m, 2*m), snd(2*m, 2*m), dsave(29)
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c ************
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c
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c Subroutine mainlb
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c
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c This subroutine solves bound constrained optimization problems by
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c using the compact formula of the limited memory BFGS updates.
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c
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c n is an integer variable.
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c On entry n is the number of variables.
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c On exit n is unchanged.
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c
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c m is an integer variable.
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c On entry m is the maximum number of variable metric
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c corrections allowed in the limited memory matrix.
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c On exit m is unchanged.
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c
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c x is a double precision array of dimension n.
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c On entry x is an approximation to the solution.
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c On exit x is the current approximation.
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c
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c l is a double precision array of dimension n.
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c On entry l is the lower bound of x.
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c On exit l is unchanged.
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c
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c u is a double precision array of dimension n.
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c On entry u is the upper bound of x.
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c On exit u is unchanged.
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c
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c nbd is an integer array of dimension n.
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c On entry nbd represents the type of bounds imposed on the
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c variables, and must be specified as follows:
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c nbd(i)=0 if x(i) is unbounded,
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c 1 if x(i) has only a lower bound,
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c 2 if x(i) has both lower and upper bounds,
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c 3 if x(i) has only an upper bound.
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c On exit nbd is unchanged.
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c
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c f is a double precision variable.
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c On first entry f is unspecified.
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c On final exit f is the value of the function at x.
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c
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c g is a double precision array of dimension n.
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c On first entry g is unspecified.
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c On final exit g is the value of the gradient at x.
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c
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c factr is a double precision variable.
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c On entry factr >= 0 is specified by the user. The iteration
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c will stop when
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c
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c (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch
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c
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c where epsmch is the machine precision, which is automatically
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c generated by the code.
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c On exit factr is unchanged.
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c
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c pgtol is a double precision variable.
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c On entry pgtol >= 0 is specified by the user. The iteration
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c will stop when
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c
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c max{|proj g_i | i = 1, ..., n} <= pgtol
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c
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c where pg_i is the ith component of the projected gradient.
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c On exit pgtol is unchanged.
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c
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c ws, wy, sy, and wt are double precision working arrays used to
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c store the following information defining the limited memory
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c BFGS matrix:
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c ws, of dimension n x m, stores S, the matrix of s-vectors;
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c wy, of dimension n x m, stores Y, the matrix of y-vectors;
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c sy, of dimension m x m, stores S'Y;
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c ss, of dimension m x m, stores S'S;
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c yy, of dimension m x m, stores Y'Y;
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c wt, of dimension m x m, stores the Cholesky factorization
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c of (theta*S'S+LD^(-1)L'); see eq.
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c (2.26) in [3].
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c
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c wn is a double precision working array of dimension 2m x 2m
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c used to store the LEL^T factorization of the indefinite matrix
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c K = [-D -Y'ZZ'Y/theta L_a'-R_z' ]
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c [L_a -R_z theta*S'AA'S ]
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c
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c where E = [-I 0]
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c [ 0 I]
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c
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c snd is a double precision working array of dimension 2m x 2m
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c used to store the lower triangular part of
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c N = [Y' ZZ'Y L_a'+R_z']
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c [L_a +R_z S'AA'S ]
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c
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c z(n),r(n),d(n),t(n), xp(n),wa(8*m) are double precision working arrays.
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c z is used at different times to store the Cauchy point and
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c the Newton point.
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c xp is used to safeguard the projected Newton direction
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c
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c sg(m),sgo(m),yg(m),ygo(m) are double precision working arrays.
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c
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c index is an integer working array of dimension n.
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c In subroutine freev, index is used to store the free and fixed
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c variables at the Generalized Cauchy Point (GCP).
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c
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c iwhere is an integer working array of dimension n used to record
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c the status of the vector x for GCP computation.
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c iwhere(i)=0 or -3 if x(i) is free and has bounds,
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c 1 if x(i) is fixed at l(i), and l(i) .ne. u(i)
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c 2 if x(i) is fixed at u(i), and u(i) .ne. l(i)
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c 3 if x(i) is always fixed, i.e., u(i)=x(i)=l(i)
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c -1 if x(i) is always free, i.e., no bounds on it.
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c
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c indx2 is an integer working array of dimension n.
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c Within subroutine cauchy, indx2 corresponds to the array iorder.
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c In subroutine freev, a list of variables entering and leaving
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c the free set is stored in indx2, and it is passed on to
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c subroutine formk with this information.
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c
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c task is a working string of characters of length 60 indicating
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c the current job when entering and leaving this subroutine.
|
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c
|
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c iprint is an INTEGER variable that must be set by the user.
|
|
c It controls the frequency and type of output generated:
|
|
c iprint<0 no output is generated;
|
|
c iprint=0 print only one line at the last iteration;
|
|
c 0<iprint<99 print also f and |proj g| every iprint iterations;
|
|
c iprint=99 print details of every iteration except n-vectors;
|
|
c iprint=100 print also the changes of active set and final x;
|
|
c iprint>100 print details of every iteration including x and g;
|
|
c When iprint > 0, the file iterate.dat will be created to
|
|
c summarize the iteration.
|
|
c
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|
c csave is a working string of characters of length 60.
|
|
c
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c lsave is a logical working array of dimension 4.
|
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c
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c isave is an integer working array of dimension 23.
|
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c
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|
c dsave is a double precision working array of dimension 29.
|
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c
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c
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|
c Subprograms called
|
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c
|
|
c L-BFGS-B Library ... cauchy, subsm, lnsrlb, formk,
|
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c
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|
c errclb, prn1lb, prn2lb, prn3lb, active, projgr,
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c
|
|
c freev, cmprlb, matupd, formt.
|
|
c
|
|
c Minpack2 Library ... timer
|
|
c
|
|
c Linpack Library ... dcopy, ddot.
|
|
c
|
|
c
|
|
c References:
|
|
c
|
|
c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
|
|
c memory algorithm for bound constrained optimization'',
|
|
c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.
|
|
c
|
|
c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN
|
|
c Subroutines for Large Scale Bound Constrained Optimization''
|
|
c Tech. Report, NAM-11, EECS Department, Northwestern University,
|
|
c 1994.
|
|
c
|
|
c [3] R. Byrd, J. Nocedal and R. Schnabel "Representations of
|
|
c Quasi-Newton Matrices and their use in Limited Memory Methods'',
|
|
c Mathematical Programming 63 (1994), no. 4, pp. 129-156.
|
|
c
|
|
c (Postscript files of these papers are available via anonymous
|
|
c ftp to eecs.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
logical prjctd,cnstnd,boxed,updatd,wrk
|
|
character*3 word
|
|
integer i,k,nintol,itfile,iback,nskip,
|
|
+ head,col,iter,itail,iupdat,
|
|
+ nseg,nfgv,info,ifun,
|
|
+ iword,nfree,nact,ileave,nenter
|
|
double precision theta,fold,ddot,dr,rr,tol,
|
|
+ xstep,sbgnrm,ddum,dnorm,dtd,epsmch,
|
|
+ cpu1,cpu2,cachyt,sbtime,lnscht,time1,time2,
|
|
+ gd,gdold,stp,stpmx,time
|
|
double precision one,zero
|
|
parameter (one=1.0d0,zero=0.0d0)
|
|
|
|
if (task .eq. 'START') then
|
|
|
|
epsmch = epsilon(one)
|
|
|
|
call timer(time1)
|
|
|
|
c Initialize counters and scalars when task='START'.
|
|
|
|
c for the limited memory BFGS matrices:
|
|
col = 0
|
|
head = 1
|
|
theta = one
|
|
iupdat = 0
|
|
updatd = .false.
|
|
iback = 0
|
|
itail = 0
|
|
iword = 0
|
|
nact = 0
|
|
ileave = 0
|
|
nenter = 0
|
|
fold = zero
|
|
dnorm = zero
|
|
cpu1 = zero
|
|
gd = zero
|
|
stpmx = zero
|
|
sbgnrm = zero
|
|
stp = zero
|
|
gdold = zero
|
|
dtd = zero
|
|
|
|
c for operation counts:
|
|
iter = 0
|
|
nfgv = 0
|
|
nseg = 0
|
|
nintol = 0
|
|
nskip = 0
|
|
nfree = n
|
|
ifun = 0
|
|
c for stopping tolerance:
|
|
tol = factr*epsmch
|
|
|
|
c for measuring running time:
|
|
cachyt = 0
|
|
sbtime = 0
|
|
lnscht = 0
|
|
|
|
c 'word' records the status of subspace solutions.
|
|
word = '---'
|
|
|
|
c 'info' records the termination information.
|
|
info = 0
|
|
|
|
itfile = 8
|
|
if (iprint .ge. 1) then
|
|
c open a summary file 'iterate.dat'
|
|
open (8, file = 'iterate.dat', status = 'unknown')
|
|
endif
|
|
|
|
c Check the input arguments for errors.
|
|
|
|
call errclb(n,m,factr,l,u,nbd,task,info,k)
|
|
if (task(1:5) .eq. 'ERROR') then
|
|
call prn3lb(n,x,f,task,iprint,info,itfile,
|
|
+ iter,nfgv,nintol,nskip,nact,sbgnrm,
|
|
+ zero,nseg,word,iback,stp,xstep,k,
|
|
+ cachyt,sbtime,lnscht)
|
|
return
|
|
endif
|
|
|
|
call prn1lb(n,m,l,u,x,iprint,itfile,epsmch)
|
|
|
|
c Initialize iwhere & project x onto the feasible set.
|
|
|
|
call active(n,l,u,nbd,x,iwhere,iprint,prjctd,cnstnd,boxed)
|
|
|
|
c The end of the initialization.
|
|
|
|
else
|
|
c restore local variables.
|
|
|
|
prjctd = lsave(1)
|
|
cnstnd = lsave(2)
|
|
boxed = lsave(3)
|
|
updatd = lsave(4)
|
|
|
|
nintol = isave(1)
|
|
itfile = isave(3)
|
|
iback = isave(4)
|
|
nskip = isave(5)
|
|
head = isave(6)
|
|
col = isave(7)
|
|
itail = isave(8)
|
|
iter = isave(9)
|
|
iupdat = isave(10)
|
|
nseg = isave(12)
|
|
nfgv = isave(13)
|
|
info = isave(14)
|
|
ifun = isave(15)
|
|
iword = isave(16)
|
|
nfree = isave(17)
|
|
nact = isave(18)
|
|
ileave = isave(19)
|
|
nenter = isave(20)
|
|
|
|
theta = dsave(1)
|
|
fold = dsave(2)
|
|
tol = dsave(3)
|
|
dnorm = dsave(4)
|
|
epsmch = dsave(5)
|
|
cpu1 = dsave(6)
|
|
cachyt = dsave(7)
|
|
sbtime = dsave(8)
|
|
lnscht = dsave(9)
|
|
time1 = dsave(10)
|
|
gd = dsave(11)
|
|
stpmx = dsave(12)
|
|
sbgnrm = dsave(13)
|
|
stp = dsave(14)
|
|
gdold = dsave(15)
|
|
dtd = dsave(16)
|
|
|
|
c After returning from the driver go to the point where execution
|
|
c is to resume.
|
|
|
|
if (task(1:5) .eq. 'FG_LN') goto 666
|
|
if (task(1:5) .eq. 'NEW_X') goto 777
|
|
if (task(1:5) .eq. 'FG_ST') goto 111
|
|
if (task(1:4) .eq. 'STOP') then
|
|
if (task(7:9) .eq. 'CPU') then
|
|
c restore the previous iterate.
|
|
call dcopy(n,t,1,x,1)
|
|
call dcopy(n,r,1,g,1)
|
|
f = fold
|
|
endif
|
|
goto 999
|
|
endif
|
|
endif
|
|
|
|
c Compute f0 and g0.
|
|
|
|
task = 'FG_START'
|
|
c return to the driver to calculate f and g; reenter at 111.
|
|
goto 1000
|
|
111 continue
|
|
nfgv = 1
|
|
|
|
c Compute the infinity norm of the (-) projected gradient.
|
|
|
|
call projgr(n,l,u,nbd,x,g,sbgnrm)
|
|
|
|
if (iprint .ge. 1) then
|
|
write (6,1002) iter,f,sbgnrm
|
|
write (itfile,1003) iter,nfgv,sbgnrm,f
|
|
endif
|
|
if (sbgnrm .le. pgtol) then
|
|
c terminate the algorithm.
|
|
task = 'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL'
|
|
goto 999
|
|
endif
|
|
|
|
c ----------------- the beginning of the loop --------------------------
|
|
|
|
222 continue
|
|
if (iprint .ge. 99) write (6,1001) iter + 1
|
|
iword = -1
|
|
c
|
|
if (.not. cnstnd .and. col .gt. 0) then
|
|
c skip the search for GCP.
|
|
call dcopy(n,x,1,z,1)
|
|
wrk = updatd
|
|
nseg = 0
|
|
goto 333
|
|
endif
|
|
|
|
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
|
|
c
|
|
c Compute the Generalized Cauchy Point (GCP).
|
|
c
|
|
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
|
|
|
|
call timer(cpu1)
|
|
call cauchy(n,x,l,u,nbd,g,indx2,iwhere,t,d,z,
|
|
+ m,wy,ws,sy,wt,theta,col,head,
|
|
+ wa(1),wa(2*m+1),wa(4*m+1),wa(6*m+1),nseg,
|
|
+ iprint, sbgnrm, info, epsmch)
|
|
if (info .ne. 0) then
|
|
c singular triangular system detected; refresh the lbfgs memory.
|
|
if(iprint .ge. 1) write (6, 1005)
|
|
info = 0
|
|
col = 0
|
|
head = 1
|
|
theta = one
|
|
iupdat = 0
|
|
updatd = .false.
|
|
call timer(cpu2)
|
|
cachyt = cachyt + cpu2 - cpu1
|
|
goto 222
|
|
endif
|
|
call timer(cpu2)
|
|
cachyt = cachyt + cpu2 - cpu1
|
|
nintol = nintol + nseg
|
|
|
|
c Count the entering and leaving variables for iter > 0;
|
|
c find the index set of free and active variables at the GCP.
|
|
|
|
call freev(n,nfree,index,nenter,ileave,indx2,
|
|
+ iwhere,wrk,updatd,cnstnd,iprint,iter)
|
|
nact = n - nfree
|
|
|
|
333 continue
|
|
|
|
c If there are no free variables or B=theta*I, then
|
|
c skip the subspace minimization.
|
|
|
|
if (nfree .eq. 0 .or. col .eq. 0) goto 555
|
|
|
|
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
|
|
c
|
|
c Subspace minimization.
|
|
c
|
|
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
|
|
|
|
call timer(cpu1)
|
|
|
|
c Form the LEL^T factorization of the indefinite
|
|
c matrix K = [-D -Y'ZZ'Y/theta L_a'-R_z' ]
|
|
c [L_a -R_z theta*S'AA'S ]
|
|
c where E = [-I 0]
|
|
c [ 0 I]
|
|
|
|
if (wrk) call formk(n,nfree,index,nenter,ileave,indx2,iupdat,
|
|
+ updatd,wn,snd,m,ws,wy,sy,theta,col,head,info)
|
|
if (info .ne. 0) then
|
|
c nonpositive definiteness in Cholesky factorization;
|
|
c refresh the lbfgs memory and restart the iteration.
|
|
if(iprint .ge. 1) write (6, 1006)
|
|
info = 0
|
|
col = 0
|
|
head = 1
|
|
theta = one
|
|
iupdat = 0
|
|
updatd = .false.
|
|
call timer(cpu2)
|
|
sbtime = sbtime + cpu2 - cpu1
|
|
goto 222
|
|
endif
|
|
|
|
c compute r=-Z'B(xcp-xk)-Z'g (using wa(2m+1)=W'(xcp-x)
|
|
c from 'cauchy').
|
|
call cmprlb(n,m,x,g,ws,wy,sy,wt,z,r,wa,index,
|
|
+ theta,col,head,nfree,cnstnd,info)
|
|
if (info .ne. 0) goto 444
|
|
|
|
c-jlm-jn call the direct method.
|
|
|
|
call subsm( n, m, nfree, index, l, u, nbd, z, r, xp, ws, wy,
|
|
+ theta, x, g, col, head, iword, wa, wn, iprint, info)
|
|
444 continue
|
|
if (info .ne. 0) then
|
|
c singular triangular system detected;
|
|
c refresh the lbfgs memory and restart the iteration.
|
|
if(iprint .ge. 1) write (6, 1005)
|
|
info = 0
|
|
col = 0
|
|
head = 1
|
|
theta = one
|
|
iupdat = 0
|
|
updatd = .false.
|
|
call timer(cpu2)
|
|
sbtime = sbtime + cpu2 - cpu1
|
|
goto 222
|
|
endif
|
|
|
|
call timer(cpu2)
|
|
sbtime = sbtime + cpu2 - cpu1
|
|
555 continue
|
|
|
|
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
|
|
c
|
|
c Line search and optimality tests.
|
|
c
|
|
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
|
|
|
|
c Generate the search direction d:=z-x.
|
|
|
|
do 40 i = 1, n
|
|
d(i) = z(i) - x(i)
|
|
40 continue
|
|
call timer(cpu1)
|
|
666 continue
|
|
call lnsrlb(n,l,u,nbd,x,f,fold,gd,gdold,g,d,r,t,z,stp,dnorm,
|
|
+ dtd,xstep,stpmx,iter,ifun,iback,nfgv,info,task,
|
|
+ boxed,cnstnd,csave,isave(22),dsave(17))
|
|
if (info .ne. 0 .or. iback .ge. 20) then
|
|
c restore the previous iterate.
|
|
call dcopy(n,t,1,x,1)
|
|
call dcopy(n,r,1,g,1)
|
|
f = fold
|
|
if (col .eq. 0) then
|
|
c abnormal termination.
|
|
if (info .eq. 0) then
|
|
info = -9
|
|
c restore the actual number of f and g evaluations etc.
|
|
nfgv = nfgv - 1
|
|
ifun = ifun - 1
|
|
iback = iback - 1
|
|
endif
|
|
task = 'ABNORMAL_TERMINATION_IN_LNSRCH'
|
|
iter = iter + 1
|
|
goto 999
|
|
else
|
|
c refresh the lbfgs memory and restart the iteration.
|
|
if(iprint .ge. 1) write (6, 1008)
|
|
if (info .eq. 0) nfgv = nfgv - 1
|
|
info = 0
|
|
col = 0
|
|
head = 1
|
|
theta = one
|
|
iupdat = 0
|
|
updatd = .false.
|
|
task = 'RESTART_FROM_LNSRCH'
|
|
call timer(cpu2)
|
|
lnscht = lnscht + cpu2 - cpu1
|
|
goto 222
|
|
endif
|
|
else if (task(1:5) .eq. 'FG_LN') then
|
|
c return to the driver for calculating f and g; reenter at 666.
|
|
goto 1000
|
|
else
|
|
c calculate and print out the quantities related to the new X.
|
|
call timer(cpu2)
|
|
lnscht = lnscht + cpu2 - cpu1
|
|
iter = iter + 1
|
|
|
|
c Compute the infinity norm of the projected (-)gradient.
|
|
|
|
call projgr(n,l,u,nbd,x,g,sbgnrm)
|
|
|
|
c Print iteration information.
|
|
|
|
call prn2lb(n,x,f,g,iprint,itfile,iter,nfgv,nact,
|
|
+ sbgnrm,nseg,word,iword,iback,stp,xstep)
|
|
goto 1000
|
|
endif
|
|
777 continue
|
|
|
|
c Test for termination.
|
|
|
|
if (sbgnrm .le. pgtol) then
|
|
c terminate the algorithm.
|
|
task = 'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL'
|
|
goto 999
|
|
endif
|
|
|
|
ddum = max(abs(fold), abs(f), one)
|
|
if ((fold - f) .le. tol*ddum) then
|
|
c terminate the algorithm.
|
|
task = 'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'
|
|
if (iback .ge. 10) info = -5
|
|
c i.e., to issue a warning if iback>10 in the line search.
|
|
goto 999
|
|
endif
|
|
|
|
c Compute d=newx-oldx, r=newg-oldg, rr=y'y and dr=y's.
|
|
|
|
do 42 i = 1, n
|
|
r(i) = g(i) - r(i)
|
|
42 continue
|
|
rr = ddot(n,r,1,r,1)
|
|
if (stp .eq. one) then
|
|
dr = gd - gdold
|
|
ddum = -gdold
|
|
else
|
|
dr = (gd - gdold)*stp
|
|
call dscal(n,stp,d,1)
|
|
ddum = -gdold*stp
|
|
endif
|
|
|
|
if (dr .le. epsmch*ddum) then
|
|
c skip the L-BFGS update.
|
|
nskip = nskip + 1
|
|
updatd = .false.
|
|
if (iprint .ge. 1) write (6,1004) dr, ddum
|
|
goto 888
|
|
endif
|
|
|
|
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
|
|
c
|
|
c Update the L-BFGS matrix.
|
|
c
|
|
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
|
|
|
|
updatd = .true.
|
|
iupdat = iupdat + 1
|
|
|
|
c Update matrices WS and WY and form the middle matrix in B.
|
|
|
|
call matupd(n,m,ws,wy,sy,ss,d,r,itail,
|
|
+ iupdat,col,head,theta,rr,dr,stp,dtd)
|
|
|
|
c Form the upper half of the pds T = theta*SS + L*D^(-1)*L';
|
|
c Store T in the upper triangular of the array wt;
|
|
c Cholesky factorize T to J*J' with
|
|
c J' stored in the upper triangular of wt.
|
|
|
|
call formt(m,wt,sy,ss,col,theta,info)
|
|
|
|
if (info .ne. 0) then
|
|
c nonpositive definiteness in Cholesky factorization;
|
|
c refresh the lbfgs memory and restart the iteration.
|
|
if(iprint .ge. 1) write (6, 1007)
|
|
info = 0
|
|
col = 0
|
|
head = 1
|
|
theta = one
|
|
iupdat = 0
|
|
updatd = .false.
|
|
goto 222
|
|
endif
|
|
|
|
c Now the inverse of the middle matrix in B is
|
|
|
|
c [ D^(1/2) O ] [ -D^(1/2) D^(-1/2)*L' ]
|
|
c [ -L*D^(-1/2) J ] [ 0 J' ]
|
|
|
|
888 continue
|
|
|
|
c -------------------- the end of the loop -----------------------------
|
|
|
|
goto 222
|
|
999 continue
|
|
call timer(time2)
|
|
time = time2 - time1
|
|
call prn3lb(n,x,f,task,iprint,info,itfile,
|
|
+ iter,nfgv,nintol,nskip,nact,sbgnrm,
|
|
+ time,nseg,word,iback,stp,xstep,k,
|
|
+ cachyt,sbtime,lnscht)
|
|
1000 continue
|
|
|
|
c Save local variables.
|
|
|
|
lsave(1) = prjctd
|
|
lsave(2) = cnstnd
|
|
lsave(3) = boxed
|
|
lsave(4) = updatd
|
|
|
|
isave(1) = nintol
|
|
isave(3) = itfile
|
|
isave(4) = iback
|
|
isave(5) = nskip
|
|
isave(6) = head
|
|
isave(7) = col
|
|
isave(8) = itail
|
|
isave(9) = iter
|
|
isave(10) = iupdat
|
|
isave(12) = nseg
|
|
isave(13) = nfgv
|
|
isave(14) = info
|
|
isave(15) = ifun
|
|
isave(16) = iword
|
|
isave(17) = nfree
|
|
isave(18) = nact
|
|
isave(19) = ileave
|
|
isave(20) = nenter
|
|
|
|
dsave(1) = theta
|
|
dsave(2) = fold
|
|
dsave(3) = tol
|
|
dsave(4) = dnorm
|
|
dsave(5) = epsmch
|
|
dsave(6) = cpu1
|
|
dsave(7) = cachyt
|
|
dsave(8) = sbtime
|
|
dsave(9) = lnscht
|
|
dsave(10) = time1
|
|
dsave(11) = gd
|
|
dsave(12) = stpmx
|
|
dsave(13) = sbgnrm
|
|
dsave(14) = stp
|
|
dsave(15) = gdold
|
|
dsave(16) = dtd
|
|
|
|
1001 format (//,'ITERATION ',i5)
|
|
1002 format
|
|
+ (/,'At iterate',i5,4x,'f= ',1p,d12.5,4x,'|proj g|= ',1p,d12.5)
|
|
1003 format (2(1x,i4),5x,'-',5x,'-',3x,'-',5x,'-',5x,'-',8x,'-',3x,
|
|
+ 1p,2(1x,d10.3))
|
|
1004 format (' ys=',1p,e10.3,' -gs=',1p,e10.3,' BFGS update SKIPPED')
|
|
1005 format (/,
|
|
+' Singular triangular system detected;',/,
|
|
+' refresh the lbfgs memory and restart the iteration.')
|
|
1006 format (/,
|
|
+' Nonpositive definiteness in Cholesky factorization in formk;',/,
|
|
+' refresh the lbfgs memory and restart the iteration.')
|
|
1007 format (/,
|
|
+' Nonpositive definiteness in Cholesky factorization in formt;',/,
|
|
+' refresh the lbfgs memory and restart the iteration.')
|
|
1008 format (/,
|
|
+' Bad direction in the line search;',/,
|
|
+' refresh the lbfgs memory and restart the iteration.')
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of mainlb =============================
|
|
|
|
subroutine active(n, l, u, nbd, x, iwhere, iprint,
|
|
+ prjctd, cnstnd, boxed)
|
|
|
|
logical prjctd, cnstnd, boxed
|
|
integer n, iprint, nbd(n), iwhere(n)
|
|
double precision x(n), l(n), u(n)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine active
|
|
c
|
|
c This subroutine initializes iwhere and projects the initial x to
|
|
c the feasible set if necessary.
|
|
c
|
|
c iwhere is an integer array of dimension n.
|
|
c On entry iwhere is unspecified.
|
|
c On exit iwhere(i)=-1 if x(i) has no bounds
|
|
c 3 if l(i)=u(i)
|
|
c 0 otherwise.
|
|
c In cauchy, iwhere is given finer gradations.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer nbdd,i
|
|
double precision zero
|
|
parameter (zero=0.0d0)
|
|
|
|
c Initialize nbdd, prjctd, cnstnd and boxed.
|
|
|
|
nbdd = 0
|
|
prjctd = .false.
|
|
cnstnd = .false.
|
|
boxed = .true.
|
|
|
|
c Project the initial x to the easible set if necessary.
|
|
|
|
do 10 i = 1, n
|
|
if (nbd(i) .gt. 0) then
|
|
if (nbd(i) .le. 2 .and. x(i) .le. l(i)) then
|
|
if (x(i) .lt. l(i)) then
|
|
prjctd = .true.
|
|
x(i) = l(i)
|
|
endif
|
|
nbdd = nbdd + 1
|
|
else if (nbd(i) .ge. 2 .and. x(i) .ge. u(i)) then
|
|
if (x(i) .gt. u(i)) then
|
|
prjctd = .true.
|
|
x(i) = u(i)
|
|
endif
|
|
nbdd = nbdd + 1
|
|
endif
|
|
endif
|
|
10 continue
|
|
|
|
c Initialize iwhere and assign values to cnstnd and boxed.
|
|
|
|
do 20 i = 1, n
|
|
if (nbd(i) .ne. 2) boxed = .false.
|
|
if (nbd(i) .eq. 0) then
|
|
c this variable is always free
|
|
iwhere(i) = -1
|
|
|
|
c otherwise set x(i)=mid(x(i), u(i), l(i)).
|
|
else
|
|
cnstnd = .true.
|
|
if (nbd(i) .eq. 2 .and. u(i) - l(i) .le. zero) then
|
|
c this variable is always fixed
|
|
iwhere(i) = 3
|
|
else
|
|
iwhere(i) = 0
|
|
endif
|
|
endif
|
|
20 continue
|
|
|
|
if (iprint .ge. 0) then
|
|
if (prjctd) write (6,*)
|
|
+ 'The initial X is infeasible. Restart with its projection.'
|
|
if (.not. cnstnd)
|
|
+ write (6,*) 'This problem is unconstrained.'
|
|
endif
|
|
|
|
if (iprint .gt. 0) write (6,1001) nbdd
|
|
|
|
1001 format (/,'At X0 ',i9,' variables are exactly at the bounds')
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of active =============================
|
|
|
|
subroutine bmv(m, sy, wt, col, v, p, info)
|
|
|
|
integer m, col, info
|
|
double precision sy(m, m), wt(m, m), v(2*col), p(2*col)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine bmv
|
|
c
|
|
c This subroutine computes the product of the 2m x 2m middle matrix
|
|
c in the compact L-BFGS formula of B and a 2m vector v;
|
|
c it returns the product in p.
|
|
c
|
|
c m is an integer variable.
|
|
c On entry m is the maximum number of variable metric corrections
|
|
c used to define the limited memory matrix.
|
|
c On exit m is unchanged.
|
|
c
|
|
c sy is a double precision array of dimension m x m.
|
|
c On entry sy specifies the matrix S'Y.
|
|
c On exit sy is unchanged.
|
|
c
|
|
c wt is a double precision array of dimension m x m.
|
|
c On entry wt specifies the upper triangular matrix J' which is
|
|
c the Cholesky factor of (thetaS'S+LD^(-1)L').
|
|
c On exit wt is unchanged.
|
|
c
|
|
c col is an integer variable.
|
|
c On entry col specifies the number of s-vectors (or y-vectors)
|
|
c stored in the compact L-BFGS formula.
|
|
c On exit col is unchanged.
|
|
c
|
|
c v is a double precision array of dimension 2col.
|
|
c On entry v specifies vector v.
|
|
c On exit v is unchanged.
|
|
c
|
|
c p is a double precision array of dimension 2col.
|
|
c On entry p is unspecified.
|
|
c On exit p is the product Mv.
|
|
c
|
|
c info is an integer variable.
|
|
c On entry info is unspecified.
|
|
c On exit info = 0 for normal return,
|
|
c = nonzero for abnormal return when the system
|
|
c to be solved by dtrsl is singular.
|
|
c
|
|
c Subprograms called:
|
|
c
|
|
c Linpack ... dtrsl.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer i,k,i2
|
|
double precision sum
|
|
|
|
if (col .eq. 0) return
|
|
|
|
c PART I: solve [ D^(1/2) O ] [ p1 ] = [ v1 ]
|
|
c [ -L*D^(-1/2) J ] [ p2 ] [ v2 ].
|
|
|
|
c solve Jp2=v2+LD^(-1)v1.
|
|
p(col + 1) = v(col + 1)
|
|
do 20 i = 2, col
|
|
i2 = col + i
|
|
sum = 0.0d0
|
|
do 10 k = 1, i - 1
|
|
sum = sum + sy(i,k)*v(k)/sy(k,k)
|
|
10 continue
|
|
p(i2) = v(i2) + sum
|
|
20 continue
|
|
c Solve the triangular system
|
|
call dtrsl(wt,m,col,p(col+1),11,info)
|
|
if (info .ne. 0) return
|
|
|
|
c solve D^(1/2)p1=v1.
|
|
do 30 i = 1, col
|
|
p(i) = v(i)/sqrt(sy(i,i))
|
|
30 continue
|
|
|
|
c PART II: solve [ -D^(1/2) D^(-1/2)*L' ] [ p1 ] = [ p1 ]
|
|
c [ 0 J' ] [ p2 ] [ p2 ].
|
|
|
|
c solve J^Tp2=p2.
|
|
call dtrsl(wt,m,col,p(col+1),01,info)
|
|
if (info .ne. 0) return
|
|
|
|
c compute p1=-D^(-1/2)(p1-D^(-1/2)L'p2)
|
|
c =-D^(-1/2)p1+D^(-1)L'p2.
|
|
do 40 i = 1, col
|
|
p(i) = -p(i)/sqrt(sy(i,i))
|
|
40 continue
|
|
do 60 i = 1, col
|
|
sum = 0.d0
|
|
do 50 k = i + 1, col
|
|
sum = sum + sy(k,i)*p(col+k)/sy(i,i)
|
|
50 continue
|
|
p(i) = p(i) + sum
|
|
60 continue
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================== The end of bmv ===============================
|
|
|
|
subroutine cauchy(n, x, l, u, nbd, g, iorder, iwhere, t, d, xcp,
|
|
+ m, wy, ws, sy, wt, theta, col, head, p, c, wbp,
|
|
+ v, nseg, iprint, sbgnrm, info, epsmch)
|
|
implicit none
|
|
integer n, m, head, col, nseg, iprint, info,
|
|
+ nbd(n), iorder(n), iwhere(n)
|
|
double precision theta, epsmch,
|
|
+ x(n), l(n), u(n), g(n), t(n), d(n), xcp(n),
|
|
+ wy(n, col), ws(n, col), sy(m, m),
|
|
+ wt(m, m), p(2*m), c(2*m), wbp(2*m), v(2*m)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine cauchy
|
|
c
|
|
c For given x, l, u, g (with sbgnrm > 0), and a limited memory
|
|
c BFGS matrix B defined in terms of matrices WY, WS, WT, and
|
|
c scalars head, col, and theta, this subroutine computes the
|
|
c generalized Cauchy point (GCP), defined as the first local
|
|
c minimizer of the quadratic
|
|
c
|
|
c Q(x + s) = g's + 1/2 s'Bs
|
|
c
|
|
c along the projected gradient direction P(x-tg,l,u).
|
|
c The routine returns the GCP in xcp.
|
|
c
|
|
c n is an integer variable.
|
|
c On entry n is the dimension of the problem.
|
|
c On exit n is unchanged.
|
|
c
|
|
c x is a double precision array of dimension n.
|
|
c On entry x is the starting point for the GCP computation.
|
|
c On exit x is unchanged.
|
|
c
|
|
c l is a double precision array of dimension n.
|
|
c On entry l is the lower bound of x.
|
|
c On exit l is unchanged.
|
|
c
|
|
c u is a double precision array of dimension n.
|
|
c On entry u is the upper bound of x.
|
|
c On exit u is unchanged.
|
|
c
|
|
c nbd is an integer array of dimension n.
|
|
c On entry nbd represents the type of bounds imposed on the
|
|
c variables, and must be specified as follows:
|
|
c nbd(i)=0 if x(i) is unbounded,
|
|
c 1 if x(i) has only a lower bound,
|
|
c 2 if x(i) has both lower and upper bounds, and
|
|
c 3 if x(i) has only an upper bound.
|
|
c On exit nbd is unchanged.
|
|
c
|
|
c g is a double precision array of dimension n.
|
|
c On entry g is the gradient of f(x). g must be a nonzero vector.
|
|
c On exit g is unchanged.
|
|
c
|
|
c iorder is an integer working array of dimension n.
|
|
c iorder will be used to store the breakpoints in the piecewise
|
|
c linear path and free variables encountered. On exit,
|
|
c iorder(1),...,iorder(nleft) are indices of breakpoints
|
|
c which have not been encountered;
|
|
c iorder(nleft+1),...,iorder(nbreak) are indices of
|
|
c encountered breakpoints; and
|
|
c iorder(nfree),...,iorder(n) are indices of variables which
|
|
c have no bound constraits along the search direction.
|
|
c
|
|
c iwhere is an integer array of dimension n.
|
|
c On entry iwhere indicates only the permanently fixed (iwhere=3)
|
|
c or free (iwhere= -1) components of x.
|
|
c On exit iwhere records the status of the current x variables.
|
|
c iwhere(i)=-3 if x(i) is free and has bounds, but is not moved
|
|
c 0 if x(i) is free and has bounds, and is moved
|
|
c 1 if x(i) is fixed at l(i), and l(i) .ne. u(i)
|
|
c 2 if x(i) is fixed at u(i), and u(i) .ne. l(i)
|
|
c 3 if x(i) is always fixed, i.e., u(i)=x(i)=l(i)
|
|
c -1 if x(i) is always free, i.e., it has no bounds.
|
|
c
|
|
c t is a double precision working array of dimension n.
|
|
c t will be used to store the break points.
|
|
c
|
|
c d is a double precision array of dimension n used to store
|
|
c the Cauchy direction P(x-tg)-x.
|
|
c
|
|
c xcp is a double precision array of dimension n used to return the
|
|
c GCP on exit.
|
|
c
|
|
c m is an integer variable.
|
|
c On entry m is the maximum number of variable metric corrections
|
|
c used to define the limited memory matrix.
|
|
c On exit m is unchanged.
|
|
c
|
|
c ws, wy, sy, and wt are double precision arrays.
|
|
c On entry they store information that defines the
|
|
c limited memory BFGS matrix:
|
|
c ws(n,m) stores S, a set of s-vectors;
|
|
c wy(n,m) stores Y, a set of y-vectors;
|
|
c sy(m,m) stores S'Y;
|
|
c wt(m,m) stores the
|
|
c Cholesky factorization of (theta*S'S+LD^(-1)L').
|
|
c On exit these arrays are unchanged.
|
|
c
|
|
c theta is a double precision variable.
|
|
c On entry theta is the scaling factor specifying B_0 = theta I.
|
|
c On exit theta is unchanged.
|
|
c
|
|
c col is an integer variable.
|
|
c On entry col is the actual number of variable metric
|
|
c corrections stored so far.
|
|
c On exit col is unchanged.
|
|
c
|
|
c head is an integer variable.
|
|
c On entry head is the location of the first s-vector (or y-vector)
|
|
c in S (or Y).
|
|
c On exit col is unchanged.
|
|
c
|
|
c p is a double precision working array of dimension 2m.
|
|
c p will be used to store the vector p = W^(T)d.
|
|
c
|
|
c c is a double precision working array of dimension 2m.
|
|
c c will be used to store the vector c = W^(T)(xcp-x).
|
|
c
|
|
c wbp is a double precision working array of dimension 2m.
|
|
c wbp will be used to store the row of W corresponding
|
|
c to a breakpoint.
|
|
c
|
|
c v is a double precision working array of dimension 2m.
|
|
c
|
|
c nseg is an integer variable.
|
|
c On exit nseg records the number of quadratic segments explored
|
|
c in searching for the GCP.
|
|
c
|
|
c sg and yg are double precision arrays of dimension m.
|
|
c On entry sg and yg store S'g and Y'g correspondingly.
|
|
c On exit they are unchanged.
|
|
c
|
|
c iprint is an INTEGER variable that must be set by the user.
|
|
c It controls the frequency and type of output generated:
|
|
c iprint<0 no output is generated;
|
|
c iprint=0 print only one line at the last iteration;
|
|
c 0<iprint<99 print also f and |proj g| every iprint iterations;
|
|
c iprint=99 print details of every iteration except n-vectors;
|
|
c iprint=100 print also the changes of active set and final x;
|
|
c iprint>100 print details of every iteration including x and g;
|
|
c When iprint > 0, the file iterate.dat will be created to
|
|
c summarize the iteration.
|
|
c
|
|
c sbgnrm is a double precision variable.
|
|
c On entry sbgnrm is the norm of the projected gradient at x.
|
|
c On exit sbgnrm is unchanged.
|
|
c
|
|
c info is an integer variable.
|
|
c On entry info is 0.
|
|
c On exit info = 0 for normal return,
|
|
c = nonzero for abnormal return when the the system
|
|
c used in routine bmv is singular.
|
|
c
|
|
c Subprograms called:
|
|
c
|
|
c L-BFGS-B Library ... hpsolb, bmv.
|
|
c
|
|
c Linpack ... dscal dcopy, daxpy.
|
|
c
|
|
c
|
|
c References:
|
|
c
|
|
c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
|
|
c memory algorithm for bound constrained optimization'',
|
|
c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.
|
|
c
|
|
c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN
|
|
c Subroutines for Large Scale Bound Constrained Optimization''
|
|
c Tech. Report, NAM-11, EECS Department, Northwestern University,
|
|
c 1994.
|
|
c
|
|
c (Postscript files of these papers are available via anonymous
|
|
c ftp to eecs.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
logical xlower,xupper,bnded
|
|
integer i,j,col2,nfree,nbreak,pointr,
|
|
+ ibp,nleft,ibkmin,iter
|
|
double precision f1,f2,dt,dtm,tsum,dibp,zibp,dibp2,bkmin,
|
|
+ tu,tl,wmc,wmp,wmw,ddot,tj,tj0,neggi,sbgnrm,
|
|
+ f2_org
|
|
double precision one,zero
|
|
parameter (one=1.0d0,zero=0.0d0)
|
|
|
|
c Check the status of the variables, reset iwhere(i) if necessary;
|
|
c compute the Cauchy direction d and the breakpoints t; initialize
|
|
c the derivative f1 and the vector p = W'd (for theta = 1).
|
|
|
|
if (sbgnrm .le. zero) then
|
|
if (iprint .ge. 0) write (6,*) 'Subgnorm = 0. GCP = X.'
|
|
call dcopy(n,x,1,xcp,1)
|
|
return
|
|
endif
|
|
bnded = .true.
|
|
nfree = n + 1
|
|
nbreak = 0
|
|
ibkmin = 0
|
|
bkmin = zero
|
|
col2 = 2*col
|
|
f1 = zero
|
|
if (iprint .ge. 99) write (6,3010)
|
|
|
|
c We set p to zero and build it up as we determine d.
|
|
|
|
do 20 i = 1, col2
|
|
p(i) = zero
|
|
20 continue
|
|
|
|
c In the following loop we determine for each variable its bound
|
|
c status and its breakpoint, and update p accordingly.
|
|
c Smallest breakpoint is identified.
|
|
|
|
do 50 i = 1, n
|
|
neggi = -g(i)
|
|
if (iwhere(i) .ne. 3 .and. iwhere(i) .ne. -1) then
|
|
c if x(i) is not a constant and has bounds,
|
|
c compute the difference between x(i) and its bounds.
|
|
if (nbd(i) .le. 2) tl = x(i) - l(i)
|
|
if (nbd(i) .ge. 2) tu = u(i) - x(i)
|
|
|
|
c If a variable is close enough to a bound
|
|
c we treat it as at bound.
|
|
xlower = nbd(i) .le. 2 .and. tl .le. zero
|
|
xupper = nbd(i) .ge. 2 .and. tu .le. zero
|
|
|
|
c reset iwhere(i).
|
|
iwhere(i) = 0
|
|
if (xlower) then
|
|
if (neggi .le. zero) iwhere(i) = 1
|
|
else if (xupper) then
|
|
if (neggi .ge. zero) iwhere(i) = 2
|
|
else
|
|
if (abs(neggi) .le. zero) iwhere(i) = -3
|
|
endif
|
|
endif
|
|
pointr = head
|
|
if (iwhere(i) .ne. 0 .and. iwhere(i) .ne. -1) then
|
|
d(i) = zero
|
|
else
|
|
d(i) = neggi
|
|
f1 = f1 - neggi*neggi
|
|
c calculate p := p - W'e_i* (g_i).
|
|
do 40 j = 1, col
|
|
p(j) = p(j) + wy(i,pointr)* neggi
|
|
p(col + j) = p(col + j) + ws(i,pointr)*neggi
|
|
pointr = mod(pointr,m) + 1
|
|
40 continue
|
|
if (nbd(i) .le. 2 .and. nbd(i) .ne. 0
|
|
+ .and. neggi .lt. zero) then
|
|
c x(i) + d(i) is bounded; compute t(i).
|
|
nbreak = nbreak + 1
|
|
iorder(nbreak) = i
|
|
t(nbreak) = tl/(-neggi)
|
|
if (nbreak .eq. 1 .or. t(nbreak) .lt. bkmin) then
|
|
bkmin = t(nbreak)
|
|
ibkmin = nbreak
|
|
endif
|
|
else if (nbd(i) .ge. 2 .and. neggi .gt. zero) then
|
|
c x(i) + d(i) is bounded; compute t(i).
|
|
nbreak = nbreak + 1
|
|
iorder(nbreak) = i
|
|
t(nbreak) = tu/neggi
|
|
if (nbreak .eq. 1 .or. t(nbreak) .lt. bkmin) then
|
|
bkmin = t(nbreak)
|
|
ibkmin = nbreak
|
|
endif
|
|
else
|
|
c x(i) + d(i) is not bounded.
|
|
nfree = nfree - 1
|
|
iorder(nfree) = i
|
|
if (abs(neggi) .gt. zero) bnded = .false.
|
|
endif
|
|
endif
|
|
50 continue
|
|
|
|
c The indices of the nonzero components of d are now stored
|
|
c in iorder(1),...,iorder(nbreak) and iorder(nfree),...,iorder(n).
|
|
c The smallest of the nbreak breakpoints is in t(ibkmin)=bkmin.
|
|
|
|
if (theta .ne. one) then
|
|
c complete the initialization of p for theta not= one.
|
|
call dscal(col,theta,p(col+1),1)
|
|
endif
|
|
|
|
c Initialize GCP xcp = x.
|
|
|
|
call dcopy(n,x,1,xcp,1)
|
|
|
|
if (nbreak .eq. 0 .and. nfree .eq. n + 1) then
|
|
c is a zero vector, return with the initial xcp as GCP.
|
|
if (iprint .gt. 100) write (6,1010) (xcp(i), i = 1, n)
|
|
return
|
|
endif
|
|
|
|
c Initialize c = W'(xcp - x) = 0.
|
|
|
|
do 60 j = 1, col2
|
|
c(j) = zero
|
|
60 continue
|
|
|
|
c Initialize derivative f2.
|
|
|
|
f2 = -theta*f1
|
|
f2_org = f2
|
|
if (col .gt. 0) then
|
|
call bmv(m,sy,wt,col,p,v,info)
|
|
if (info .ne. 0) return
|
|
f2 = f2 - ddot(col2,v,1,p,1)
|
|
endif
|
|
dtm = -f1/f2
|
|
tsum = zero
|
|
nseg = 1
|
|
if (iprint .ge. 99)
|
|
+ write (6,*) 'There are ',nbreak,' breakpoints '
|
|
|
|
c If there are no breakpoints, locate the GCP and return.
|
|
|
|
if (nbreak .eq. 0) goto 888
|
|
|
|
nleft = nbreak
|
|
iter = 1
|
|
|
|
|
|
tj = zero
|
|
|
|
c------------------- the beginning of the loop -------------------------
|
|
|
|
777 continue
|
|
|
|
c Find the next smallest breakpoint;
|
|
c compute dt = t(nleft) - t(nleft + 1).
|
|
|
|
tj0 = tj
|
|
if (iter .eq. 1) then
|
|
c Since we already have the smallest breakpoint we need not do
|
|
c heapsort yet. Often only one breakpoint is used and the
|
|
c cost of heapsort is avoided.
|
|
tj = bkmin
|
|
ibp = iorder(ibkmin)
|
|
else
|
|
if (iter .eq. 2) then
|
|
c Replace the already used smallest breakpoint with the
|
|
c breakpoint numbered nbreak > nlast, before heapsort call.
|
|
if (ibkmin .ne. nbreak) then
|
|
t(ibkmin) = t(nbreak)
|
|
iorder(ibkmin) = iorder(nbreak)
|
|
endif
|
|
c Update heap structure of breakpoints
|
|
c (if iter=2, initialize heap).
|
|
endif
|
|
call hpsolb(nleft,t,iorder,iter-2)
|
|
tj = t(nleft)
|
|
ibp = iorder(nleft)
|
|
endif
|
|
|
|
dt = tj - tj0
|
|
|
|
if (dt .ne. zero .and. iprint .ge. 100) then
|
|
write (6,4011) nseg,f1,f2
|
|
write (6,5010) dt
|
|
write (6,6010) dtm
|
|
endif
|
|
|
|
c If a minimizer is within this interval, locate the GCP and return.
|
|
|
|
if (dtm .lt. dt) goto 888
|
|
|
|
c Otherwise fix one variable and
|
|
c reset the corresponding component of d to zero.
|
|
|
|
tsum = tsum + dt
|
|
nleft = nleft - 1
|
|
iter = iter + 1
|
|
dibp = d(ibp)
|
|
d(ibp) = zero
|
|
if (dibp .gt. zero) then
|
|
zibp = u(ibp) - x(ibp)
|
|
xcp(ibp) = u(ibp)
|
|
iwhere(ibp) = 2
|
|
else
|
|
zibp = l(ibp) - x(ibp)
|
|
xcp(ibp) = l(ibp)
|
|
iwhere(ibp) = 1
|
|
endif
|
|
if (iprint .ge. 100) write (6,*) 'Variable ',ibp,' is fixed.'
|
|
if (nleft .eq. 0 .and. nbreak .eq. n) then
|
|
c all n variables are fixed,
|
|
c return with xcp as GCP.
|
|
dtm = dt
|
|
goto 999
|
|
endif
|
|
|
|
c Update the derivative information.
|
|
|
|
nseg = nseg + 1
|
|
dibp2 = dibp**2
|
|
|
|
c Update f1 and f2.
|
|
|
|
c temporarily set f1 and f2 for col=0.
|
|
f1 = f1 + dt*f2 + dibp2 - theta*dibp*zibp
|
|
f2 = f2 - theta*dibp2
|
|
|
|
if (col .gt. 0) then
|
|
c update c = c + dt*p.
|
|
call daxpy(col2,dt,p,1,c,1)
|
|
|
|
c choose wbp,
|
|
c the row of W corresponding to the breakpoint encountered.
|
|
pointr = head
|
|
do 70 j = 1,col
|
|
wbp(j) = wy(ibp,pointr)
|
|
wbp(col + j) = theta*ws(ibp,pointr)
|
|
pointr = mod(pointr,m) + 1
|
|
70 continue
|
|
|
|
c compute (wbp)Mc, (wbp)Mp, and (wbp)M(wbp)'.
|
|
call bmv(m,sy,wt,col,wbp,v,info)
|
|
if (info .ne. 0) return
|
|
wmc = ddot(col2,c,1,v,1)
|
|
wmp = ddot(col2,p,1,v,1)
|
|
wmw = ddot(col2,wbp,1,v,1)
|
|
|
|
c update p = p - dibp*wbp.
|
|
call daxpy(col2,-dibp,wbp,1,p,1)
|
|
|
|
c complete updating f1 and f2 while col > 0.
|
|
f1 = f1 + dibp*wmc
|
|
f2 = f2 + 2.0d0*dibp*wmp - dibp2*wmw
|
|
endif
|
|
|
|
f2 = max(epsmch*f2_org,f2)
|
|
if (nleft .gt. 0) then
|
|
dtm = -f1/f2
|
|
goto 777
|
|
c to repeat the loop for unsearched intervals.
|
|
else if(bnded) then
|
|
f1 = zero
|
|
f2 = zero
|
|
dtm = zero
|
|
else
|
|
dtm = -f1/f2
|
|
endif
|
|
|
|
c------------------- the end of the loop -------------------------------
|
|
|
|
888 continue
|
|
if (iprint .ge. 99) then
|
|
write (6,*)
|
|
write (6,*) 'GCP found in this segment'
|
|
write (6,4010) nseg,f1,f2
|
|
write (6,6010) dtm
|
|
endif
|
|
if (dtm .le. zero) dtm = zero
|
|
tsum = tsum + dtm
|
|
|
|
c Move free variables (i.e., the ones w/o breakpoints) and
|
|
c the variables whose breakpoints haven't been reached.
|
|
|
|
call daxpy(n,tsum,d,1,xcp,1)
|
|
|
|
999 continue
|
|
|
|
c Update c = c + dtm*p = W'(x^c - x)
|
|
c which will be used in computing r = Z'(B(x^c - x) + g).
|
|
|
|
if (col .gt. 0) call daxpy(col2,dtm,p,1,c,1)
|
|
if (iprint .gt. 100) write (6,1010) (xcp(i),i = 1,n)
|
|
if (iprint .ge. 99) write (6,2010)
|
|
|
|
1010 format ('Cauchy X = ',/,(4x,1p,6(1x,d11.4)))
|
|
2010 format (/,'---------------- exit CAUCHY----------------------',/)
|
|
3010 format (/,'---------------- CAUCHY entered-------------------')
|
|
4010 format ('Piece ',i3,' --f1, f2 at start point ',1p,2(1x,d11.4))
|
|
4011 format (/,'Piece ',i3,' --f1, f2 at start point ',
|
|
+ 1p,2(1x,d11.4))
|
|
5010 format ('Distance to the next break point = ',1p,d11.4)
|
|
6010 format ('Distance to the stationary point = ',1p,d11.4)
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c====================== The end of cauchy ==============================
|
|
|
|
subroutine cmprlb(n, m, x, g, ws, wy, sy, wt, z, r, wa, index,
|
|
+ theta, col, head, nfree, cnstnd, info)
|
|
|
|
logical cnstnd
|
|
integer n, m, col, head, nfree, info, index(n)
|
|
double precision theta,
|
|
+ x(n), g(n), z(n), r(n), wa(4*m),
|
|
+ ws(n, m), wy(n, m), sy(m, m), wt(m, m)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine cmprlb
|
|
c
|
|
c This subroutine computes r=-Z'B(xcp-xk)-Z'g by using
|
|
c wa(2m+1)=W'(xcp-x) from subroutine cauchy.
|
|
c
|
|
c Subprograms called:
|
|
c
|
|
c L-BFGS-B Library ... bmv.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer i,j,k,pointr
|
|
double precision a1,a2
|
|
|
|
if (.not. cnstnd .and. col .gt. 0) then
|
|
do 26 i = 1, n
|
|
r(i) = -g(i)
|
|
26 continue
|
|
else
|
|
do 30 i = 1, nfree
|
|
k = index(i)
|
|
r(i) = -theta*(z(k) - x(k)) - g(k)
|
|
30 continue
|
|
call bmv(m,sy,wt,col,wa(2*m+1),wa(1),info)
|
|
if (info .ne. 0) then
|
|
info = -8
|
|
return
|
|
endif
|
|
pointr = head
|
|
do 34 j = 1, col
|
|
a1 = wa(j)
|
|
a2 = theta*wa(col + j)
|
|
do 32 i = 1, nfree
|
|
k = index(i)
|
|
r(i) = r(i) + wy(k,pointr)*a1 + ws(k,pointr)*a2
|
|
32 continue
|
|
pointr = mod(pointr,m) + 1
|
|
34 continue
|
|
endif
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of cmprlb =============================
|
|
|
|
subroutine errclb(n, m, factr, l, u, nbd, task, info, k)
|
|
|
|
character*60 task
|
|
integer n, m, info, k, nbd(n)
|
|
double precision factr, l(n), u(n)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine errclb
|
|
c
|
|
c This subroutine checks the validity of the input data.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer i
|
|
double precision one,zero
|
|
parameter (one=1.0d0,zero=0.0d0)
|
|
|
|
c Check the input arguments for errors.
|
|
|
|
if (n .le. 0) task = 'ERROR: N .LE. 0'
|
|
if (m .le. 0) task = 'ERROR: M .LE. 0'
|
|
if (factr .lt. zero) task = 'ERROR: FACTR .LT. 0'
|
|
|
|
c Check the validity of the arrays nbd(i), u(i), and l(i).
|
|
|
|
do 10 i = 1, n
|
|
if (nbd(i) .lt. 0 .or. nbd(i) .gt. 3) then
|
|
c return
|
|
task = 'ERROR: INVALID NBD'
|
|
info = -6
|
|
k = i
|
|
endif
|
|
if (nbd(i) .eq. 2) then
|
|
if (l(i) .gt. u(i)) then
|
|
c return
|
|
task = 'ERROR: NO FEASIBLE SOLUTION'
|
|
info = -7
|
|
k = i
|
|
endif
|
|
endif
|
|
10 continue
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of errclb =============================
|
|
|
|
subroutine formk(n, nsub, ind, nenter, ileave, indx2, iupdat,
|
|
+ updatd, wn, wn1, m, ws, wy, sy, theta, col,
|
|
+ head, info)
|
|
|
|
integer n, nsub, m, col, head, nenter, ileave, iupdat,
|
|
+ info, ind(n), indx2(n)
|
|
double precision theta, wn(2*m, 2*m), wn1(2*m, 2*m),
|
|
+ ws(n, m), wy(n, m), sy(m, m)
|
|
logical updatd
|
|
|
|
c ************
|
|
c
|
|
c Subroutine formk
|
|
c
|
|
c This subroutine forms the LEL^T factorization of the indefinite
|
|
c
|
|
c matrix K = [-D -Y'ZZ'Y/theta L_a'-R_z' ]
|
|
c [L_a -R_z theta*S'AA'S ]
|
|
c where E = [-I 0]
|
|
c [ 0 I]
|
|
c The matrix K can be shown to be equal to the matrix M^[-1]N
|
|
c occurring in section 5.1 of [1], as well as to the matrix
|
|
c Mbar^[-1] Nbar in section 5.3.
|
|
c
|
|
c n is an integer variable.
|
|
c On entry n is the dimension of the problem.
|
|
c On exit n is unchanged.
|
|
c
|
|
c nsub is an integer variable
|
|
c On entry nsub is the number of subspace variables in free set.
|
|
c On exit nsub is not changed.
|
|
c
|
|
c ind is an integer array of dimension nsub.
|
|
c On entry ind specifies the indices of subspace variables.
|
|
c On exit ind is unchanged.
|
|
c
|
|
c nenter is an integer variable.
|
|
c On entry nenter is the number of variables entering the
|
|
c free set.
|
|
c On exit nenter is unchanged.
|
|
c
|
|
c ileave is an integer variable.
|
|
c On entry indx2(ileave),...,indx2(n) are the variables leaving
|
|
c the free set.
|
|
c On exit ileave is unchanged.
|
|
c
|
|
c indx2 is an integer array of dimension n.
|
|
c On entry indx2(1),...,indx2(nenter) are the variables entering
|
|
c the free set, while indx2(ileave),...,indx2(n) are the
|
|
c variables leaving the free set.
|
|
c On exit indx2 is unchanged.
|
|
c
|
|
c iupdat is an integer variable.
|
|
c On entry iupdat is the total number of BFGS updates made so far.
|
|
c On exit iupdat is unchanged.
|
|
c
|
|
c updatd is a logical variable.
|
|
c On entry 'updatd' is true if the L-BFGS matrix is updatd.
|
|
c On exit 'updatd' is unchanged.
|
|
c
|
|
c wn is a double precision array of dimension 2m x 2m.
|
|
c On entry wn is unspecified.
|
|
c On exit the upper triangle of wn stores the LEL^T factorization
|
|
c of the 2*col x 2*col indefinite matrix
|
|
c [-D -Y'ZZ'Y/theta L_a'-R_z' ]
|
|
c [L_a -R_z theta*S'AA'S ]
|
|
c
|
|
c wn1 is a double precision array of dimension 2m x 2m.
|
|
c On entry wn1 stores the lower triangular part of
|
|
c [Y' ZZ'Y L_a'+R_z']
|
|
c [L_a+R_z S'AA'S ]
|
|
c in the previous iteration.
|
|
c On exit wn1 stores the corresponding updated matrices.
|
|
c The purpose of wn1 is just to store these inner products
|
|
c so they can be easily updated and inserted into wn.
|
|
c
|
|
c m is an integer variable.
|
|
c On entry m is the maximum number of variable metric corrections
|
|
c used to define the limited memory matrix.
|
|
c On exit m is unchanged.
|
|
c
|
|
c ws, wy, sy, and wtyy are double precision arrays;
|
|
c theta is a double precision variable;
|
|
c col is an integer variable;
|
|
c head is an integer variable.
|
|
c On entry they store the information defining the
|
|
c limited memory BFGS matrix:
|
|
c ws(n,m) stores S, a set of s-vectors;
|
|
c wy(n,m) stores Y, a set of y-vectors;
|
|
c sy(m,m) stores S'Y;
|
|
c wtyy(m,m) stores the Cholesky factorization
|
|
c of (theta*S'S+LD^(-1)L')
|
|
c theta is the scaling factor specifying B_0 = theta I;
|
|
c col is the number of variable metric corrections stored;
|
|
c head is the location of the 1st s- (or y-) vector in S (or Y).
|
|
c On exit they are unchanged.
|
|
c
|
|
c info is an integer variable.
|
|
c On entry info is unspecified.
|
|
c On exit info = 0 for normal return;
|
|
c = -1 when the 1st Cholesky factorization failed;
|
|
c = -2 when the 2st Cholesky factorization failed.
|
|
c
|
|
c Subprograms called:
|
|
c
|
|
c Linpack ... dcopy, dpofa, dtrsl.
|
|
c
|
|
c
|
|
c References:
|
|
c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
|
|
c memory algorithm for bound constrained optimization'',
|
|
c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.
|
|
c
|
|
c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: a
|
|
c limited memory FORTRAN code for solving bound constrained
|
|
c optimization problems'', Tech. Report, NAM-11, EECS Department,
|
|
c Northwestern University, 1994.
|
|
c
|
|
c (Postscript files of these papers are available via anonymous
|
|
c ftp to eecs.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer m2,ipntr,jpntr,iy,is,jy,js,is1,js1,k1,i,k,
|
|
+ col2,pbegin,pend,dbegin,dend,upcl
|
|
double precision ddot,temp1,temp2,temp3,temp4
|
|
double precision one,zero
|
|
parameter (one=1.0d0,zero=0.0d0)
|
|
|
|
c Form the lower triangular part of
|
|
c WN1 = [Y' ZZ'Y L_a'+R_z']
|
|
c [L_a+R_z S'AA'S ]
|
|
c where L_a is the strictly lower triangular part of S'AA'Y
|
|
c R_z is the upper triangular part of S'ZZ'Y.
|
|
|
|
if (updatd) then
|
|
if (iupdat .gt. m) then
|
|
c shift old part of WN1.
|
|
do 10 jy = 1, m - 1
|
|
js = m + jy
|
|
call dcopy(m-jy,wn1(jy+1,jy+1),1,wn1(jy,jy),1)
|
|
call dcopy(m-jy,wn1(js+1,js+1),1,wn1(js,js),1)
|
|
call dcopy(m-1,wn1(m+2,jy+1),1,wn1(m+1,jy),1)
|
|
10 continue
|
|
endif
|
|
|
|
c put new rows in blocks (1,1), (2,1) and (2,2).
|
|
pbegin = 1
|
|
pend = nsub
|
|
dbegin = nsub + 1
|
|
dend = n
|
|
iy = col
|
|
is = m + col
|
|
ipntr = head + col - 1
|
|
if (ipntr .gt. m) ipntr = ipntr - m
|
|
jpntr = head
|
|
do 20 jy = 1, col
|
|
js = m + jy
|
|
temp1 = zero
|
|
temp2 = zero
|
|
temp3 = zero
|
|
c compute element jy of row 'col' of Y'ZZ'Y
|
|
do 15 k = pbegin, pend
|
|
k1 = ind(k)
|
|
temp1 = temp1 + wy(k1,ipntr)*wy(k1,jpntr)
|
|
15 continue
|
|
c compute elements jy of row 'col' of L_a and S'AA'S
|
|
do 16 k = dbegin, dend
|
|
k1 = ind(k)
|
|
temp2 = temp2 + ws(k1,ipntr)*ws(k1,jpntr)
|
|
temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr)
|
|
16 continue
|
|
wn1(iy,jy) = temp1
|
|
wn1(is,js) = temp2
|
|
wn1(is,jy) = temp3
|
|
jpntr = mod(jpntr,m) + 1
|
|
20 continue
|
|
|
|
c put new column in block (2,1).
|
|
jy = col
|
|
jpntr = head + col - 1
|
|
if (jpntr .gt. m) jpntr = jpntr - m
|
|
ipntr = head
|
|
do 30 i = 1, col
|
|
is = m + i
|
|
temp3 = zero
|
|
c compute element i of column 'col' of R_z
|
|
do 25 k = pbegin, pend
|
|
k1 = ind(k)
|
|
temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr)
|
|
25 continue
|
|
ipntr = mod(ipntr,m) + 1
|
|
wn1(is,jy) = temp3
|
|
30 continue
|
|
upcl = col - 1
|
|
else
|
|
upcl = col
|
|
endif
|
|
|
|
c modify the old parts in blocks (1,1) and (2,2) due to changes
|
|
c in the set of free variables.
|
|
ipntr = head
|
|
do 45 iy = 1, upcl
|
|
is = m + iy
|
|
jpntr = head
|
|
do 40 jy = 1, iy
|
|
js = m + jy
|
|
temp1 = zero
|
|
temp2 = zero
|
|
temp3 = zero
|
|
temp4 = zero
|
|
do 35 k = 1, nenter
|
|
k1 = indx2(k)
|
|
temp1 = temp1 + wy(k1,ipntr)*wy(k1,jpntr)
|
|
temp2 = temp2 + ws(k1,ipntr)*ws(k1,jpntr)
|
|
35 continue
|
|
do 36 k = ileave, n
|
|
k1 = indx2(k)
|
|
temp3 = temp3 + wy(k1,ipntr)*wy(k1,jpntr)
|
|
temp4 = temp4 + ws(k1,ipntr)*ws(k1,jpntr)
|
|
36 continue
|
|
wn1(iy,jy) = wn1(iy,jy) + temp1 - temp3
|
|
wn1(is,js) = wn1(is,js) - temp2 + temp4
|
|
jpntr = mod(jpntr,m) + 1
|
|
40 continue
|
|
ipntr = mod(ipntr,m) + 1
|
|
45 continue
|
|
|
|
c modify the old parts in block (2,1).
|
|
ipntr = head
|
|
do 60 is = m + 1, m + upcl
|
|
jpntr = head
|
|
do 55 jy = 1, upcl
|
|
temp1 = zero
|
|
temp3 = zero
|
|
do 50 k = 1, nenter
|
|
k1 = indx2(k)
|
|
temp1 = temp1 + ws(k1,ipntr)*wy(k1,jpntr)
|
|
50 continue
|
|
do 51 k = ileave, n
|
|
k1 = indx2(k)
|
|
temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr)
|
|
51 continue
|
|
if (is .le. jy + m) then
|
|
wn1(is,jy) = wn1(is,jy) + temp1 - temp3
|
|
else
|
|
wn1(is,jy) = wn1(is,jy) - temp1 + temp3
|
|
endif
|
|
jpntr = mod(jpntr,m) + 1
|
|
55 continue
|
|
ipntr = mod(ipntr,m) + 1
|
|
60 continue
|
|
|
|
c Form the upper triangle of WN = [D+Y' ZZ'Y/theta -L_a'+R_z' ]
|
|
c [-L_a +R_z S'AA'S*theta]
|
|
|
|
m2 = 2*m
|
|
do 70 iy = 1, col
|
|
is = col + iy
|
|
is1 = m + iy
|
|
do 65 jy = 1, iy
|
|
js = col + jy
|
|
js1 = m + jy
|
|
wn(jy,iy) = wn1(iy,jy)/theta
|
|
wn(js,is) = wn1(is1,js1)*theta
|
|
65 continue
|
|
do 66 jy = 1, iy - 1
|
|
wn(jy,is) = -wn1(is1,jy)
|
|
66 continue
|
|
do 67 jy = iy, col
|
|
wn(jy,is) = wn1(is1,jy)
|
|
67 continue
|
|
wn(iy,iy) = wn(iy,iy) + sy(iy,iy)
|
|
70 continue
|
|
|
|
c Form the upper triangle of WN= [ LL' L^-1(-L_a'+R_z')]
|
|
c [(-L_a +R_z)L'^-1 S'AA'S*theta ]
|
|
|
|
c first Cholesky factor (1,1) block of wn to get LL'
|
|
c with L' stored in the upper triangle of wn.
|
|
call dpofa(wn,m2,col,info)
|
|
if (info .ne. 0) then
|
|
info = -1
|
|
return
|
|
endif
|
|
c then form L^-1(-L_a'+R_z') in the (1,2) block.
|
|
col2 = 2*col
|
|
do 71 js = col+1 ,col2
|
|
call dtrsl(wn,m2,col,wn(1,js),11,info)
|
|
71 continue
|
|
|
|
c Form S'AA'S*theta + (L^-1(-L_a'+R_z'))'L^-1(-L_a'+R_z') in the
|
|
c upper triangle of (2,2) block of wn.
|
|
|
|
|
|
do 72 is = col+1, col2
|
|
do 74 js = is, col2
|
|
wn(is,js) = wn(is,js) + ddot(col,wn(1,is),1,wn(1,js),1)
|
|
74 continue
|
|
72 continue
|
|
|
|
c Cholesky factorization of (2,2) block of wn.
|
|
|
|
call dpofa(wn(col+1,col+1),m2,col,info)
|
|
if (info .ne. 0) then
|
|
info = -2
|
|
return
|
|
endif
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of formk ==============================
|
|
|
|
subroutine formt(m, wt, sy, ss, col, theta, info)
|
|
|
|
integer m, col, info
|
|
double precision theta, wt(m, m), sy(m, m), ss(m, m)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine formt
|
|
c
|
|
c This subroutine forms the upper half of the pos. def. and symm.
|
|
c T = theta*SS + L*D^(-1)*L', stores T in the upper triangle
|
|
c of the array wt, and performs the Cholesky factorization of T
|
|
c to produce J*J', with J' stored in the upper triangle of wt.
|
|
c
|
|
c Subprograms called:
|
|
c
|
|
c Linpack ... dpofa.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer i,j,k,k1
|
|
double precision ddum
|
|
double precision zero
|
|
parameter (zero=0.0d0)
|
|
|
|
|
|
c Form the upper half of T = theta*SS + L*D^(-1)*L',
|
|
c store T in the upper triangle of the array wt.
|
|
|
|
do 52 j = 1, col
|
|
wt(1,j) = theta*ss(1,j)
|
|
52 continue
|
|
do 55 i = 2, col
|
|
do 54 j = i, col
|
|
k1 = min(i,j) - 1
|
|
ddum = zero
|
|
do 53 k = 1, k1
|
|
ddum = ddum + sy(i,k)*sy(j,k)/sy(k,k)
|
|
53 continue
|
|
wt(i,j) = ddum + theta*ss(i,j)
|
|
54 continue
|
|
55 continue
|
|
|
|
c Cholesky factorize T to J*J' with
|
|
c J' stored in the upper triangle of wt.
|
|
|
|
call dpofa(wt,m,col,info)
|
|
if (info .ne. 0) then
|
|
info = -3
|
|
endif
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of formt ==============================
|
|
|
|
subroutine freev(n, nfree, index, nenter, ileave, indx2,
|
|
+ iwhere, wrk, updatd, cnstnd, iprint, iter)
|
|
|
|
integer n, nfree, nenter, ileave, iprint, iter,
|
|
+ index(n), indx2(n), iwhere(n)
|
|
logical wrk, updatd, cnstnd
|
|
|
|
c ************
|
|
c
|
|
c Subroutine freev
|
|
c
|
|
c This subroutine counts the entering and leaving variables when
|
|
c iter > 0, and finds the index set of free and active variables
|
|
c at the GCP.
|
|
c
|
|
c cnstnd is a logical variable indicating whether bounds are present
|
|
c
|
|
c index is an integer array of dimension n
|
|
c for i=1,...,nfree, index(i) are the indices of free variables
|
|
c for i=nfree+1,...,n, index(i) are the indices of bound variables
|
|
c On entry after the first iteration, index gives
|
|
c the free variables at the previous iteration.
|
|
c On exit it gives the free variables based on the determination
|
|
c in cauchy using the array iwhere.
|
|
c
|
|
c indx2 is an integer array of dimension n
|
|
c On entry indx2 is unspecified.
|
|
c On exit with iter>0, indx2 indicates which variables
|
|
c have changed status since the previous iteration.
|
|
c For i= 1,...,nenter, indx2(i) have changed from bound to free.
|
|
c For i= ileave+1,...,n, indx2(i) have changed from free to bound.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer iact,i,k
|
|
|
|
nenter = 0
|
|
ileave = n + 1
|
|
if (iter .gt. 0 .and. cnstnd) then
|
|
c count the entering and leaving variables.
|
|
do 20 i = 1, nfree
|
|
k = index(i)
|
|
|
|
c write(6,*) ' k = index(i) ', k
|
|
c write(6,*) ' index = ', i
|
|
|
|
if (iwhere(k) .gt. 0) then
|
|
ileave = ileave - 1
|
|
indx2(ileave) = k
|
|
if (iprint .ge. 100) write (6,*)
|
|
+ 'Variable ',k,' leaves the set of free variables'
|
|
endif
|
|
20 continue
|
|
do 22 i = 1 + nfree, n
|
|
k = index(i)
|
|
if (iwhere(k) .le. 0) then
|
|
nenter = nenter + 1
|
|
indx2(nenter) = k
|
|
if (iprint .ge. 100) write (6,*)
|
|
+ 'Variable ',k,' enters the set of free variables'
|
|
endif
|
|
22 continue
|
|
if (iprint .ge. 99) write (6,*)
|
|
+ n+1-ileave,' variables leave; ',nenter,' variables enter'
|
|
endif
|
|
wrk = (ileave .lt. n+1) .or. (nenter .gt. 0) .or. updatd
|
|
|
|
c Find the index set of free and active variables at the GCP.
|
|
|
|
nfree = 0
|
|
iact = n + 1
|
|
do 24 i = 1, n
|
|
if (iwhere(i) .le. 0) then
|
|
nfree = nfree + 1
|
|
index(nfree) = i
|
|
else
|
|
iact = iact - 1
|
|
index(iact) = i
|
|
endif
|
|
24 continue
|
|
if (iprint .ge. 99) write (6,*)
|
|
+ nfree,' variables are free at GCP ',iter + 1
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of freev ==============================
|
|
|
|
subroutine hpsolb(n, t, iorder, iheap)
|
|
integer iheap, n, iorder(n)
|
|
double precision t(n)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine hpsolb
|
|
c
|
|
c This subroutine sorts out the least element of t, and puts the
|
|
c remaining elements of t in a heap.
|
|
c
|
|
c n is an integer variable.
|
|
c On entry n is the dimension of the arrays t and iorder.
|
|
c On exit n is unchanged.
|
|
c
|
|
c t is a double precision array of dimension n.
|
|
c On entry t stores the elements to be sorted,
|
|
c On exit t(n) stores the least elements of t, and t(1) to t(n-1)
|
|
c stores the remaining elements in the form of a heap.
|
|
c
|
|
c iorder is an integer array of dimension n.
|
|
c On entry iorder(i) is the index of t(i).
|
|
c On exit iorder(i) is still the index of t(i), but iorder may be
|
|
c permuted in accordance with t.
|
|
c
|
|
c iheap is an integer variable specifying the task.
|
|
c On entry iheap should be set as follows:
|
|
c iheap .eq. 0 if t(1) to t(n) is not in the form of a heap,
|
|
c iheap .ne. 0 if otherwise.
|
|
c On exit iheap is unchanged.
|
|
c
|
|
c
|
|
c References:
|
|
c Algorithm 232 of CACM (J. W. J. Williams): HEAPSORT.
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c ************
|
|
|
|
integer i,j,k,indxin,indxou
|
|
double precision ddum,out
|
|
|
|
if (iheap .eq. 0) then
|
|
|
|
c Rearrange the elements t(1) to t(n) to form a heap.
|
|
|
|
do 20 k = 2, n
|
|
ddum = t(k)
|
|
indxin = iorder(k)
|
|
|
|
c Add ddum to the heap.
|
|
i = k
|
|
10 continue
|
|
if (i.gt.1) then
|
|
j = i/2
|
|
if (ddum .lt. t(j)) then
|
|
t(i) = t(j)
|
|
iorder(i) = iorder(j)
|
|
i = j
|
|
goto 10
|
|
endif
|
|
endif
|
|
t(i) = ddum
|
|
iorder(i) = indxin
|
|
20 continue
|
|
endif
|
|
|
|
c Assign to 'out' the value of t(1), the least member of the heap,
|
|
c and rearrange the remaining members to form a heap as
|
|
c elements 1 to n-1 of t.
|
|
|
|
if (n .gt. 1) then
|
|
i = 1
|
|
out = t(1)
|
|
indxou = iorder(1)
|
|
ddum = t(n)
|
|
indxin = iorder(n)
|
|
|
|
c Restore the heap
|
|
30 continue
|
|
j = i+i
|
|
if (j .le. n-1) then
|
|
if (t(j+1) .lt. t(j)) j = j+1
|
|
if (t(j) .lt. ddum ) then
|
|
t(i) = t(j)
|
|
iorder(i) = iorder(j)
|
|
i = j
|
|
goto 30
|
|
endif
|
|
endif
|
|
t(i) = ddum
|
|
iorder(i) = indxin
|
|
|
|
c Put the least member in t(n).
|
|
|
|
t(n) = out
|
|
iorder(n) = indxou
|
|
endif
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c====================== The end of hpsolb ==============================
|
|
|
|
subroutine lnsrlb(n, l, u, nbd, x, f, fold, gd, gdold, g, d, r, t,
|
|
+ z, stp, dnorm, dtd, xstep, stpmx, iter, ifun,
|
|
+ iback, nfgv, info, task, boxed, cnstnd, csave,
|
|
+ isave, dsave)
|
|
|
|
character*60 task, csave
|
|
logical boxed, cnstnd
|
|
integer n, iter, ifun, iback, nfgv, info,
|
|
+ nbd(n), isave(2)
|
|
double precision f, fold, gd, gdold, stp, dnorm, dtd, xstep,
|
|
+ stpmx, x(n), l(n), u(n), g(n), d(n), r(n), t(n),
|
|
+ z(n), dsave(13)
|
|
c **********
|
|
c
|
|
c Subroutine lnsrlb
|
|
c
|
|
c This subroutine calls subroutine dcsrch from the Minpack2 library
|
|
c to perform the line search. Subroutine dscrch is safeguarded so
|
|
c that all trial points lie within the feasible region.
|
|
c
|
|
c Subprograms called:
|
|
c
|
|
c Minpack2 Library ... dcsrch.
|
|
c
|
|
c Linpack ... dtrsl, ddot.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c **********
|
|
|
|
integer i
|
|
double precision ddot,a1,a2
|
|
double precision one,zero,big
|
|
parameter (one=1.0d0,zero=0.0d0,big=1.0d+10)
|
|
double precision ftol,gtol,xtol
|
|
parameter (ftol=1.0d-3,gtol=0.9d0,xtol=0.1d0)
|
|
|
|
if (task(1:5) .eq. 'FG_LN') goto 556
|
|
|
|
dtd = ddot(n,d,1,d,1)
|
|
dnorm = sqrt(dtd)
|
|
|
|
c Determine the maximum step length.
|
|
|
|
stpmx = big
|
|
if (cnstnd) then
|
|
if (iter .eq. 0) then
|
|
stpmx = one
|
|
else
|
|
do 43 i = 1, n
|
|
a1 = d(i)
|
|
if (nbd(i) .ne. 0) then
|
|
if (a1 .lt. zero .and. nbd(i) .le. 2) then
|
|
a2 = l(i) - x(i)
|
|
if (a2 .ge. zero) then
|
|
stpmx = zero
|
|
else if (a1*stpmx .lt. a2) then
|
|
stpmx = a2/a1
|
|
endif
|
|
else if (a1 .gt. zero .and. nbd(i) .ge. 2) then
|
|
a2 = u(i) - x(i)
|
|
if (a2 .le. zero) then
|
|
stpmx = zero
|
|
else if (a1*stpmx .gt. a2) then
|
|
stpmx = a2/a1
|
|
endif
|
|
endif
|
|
endif
|
|
43 continue
|
|
endif
|
|
endif
|
|
|
|
if (iter .eq. 0 .and. .not. boxed) then
|
|
stp = min(one/dnorm, stpmx)
|
|
else
|
|
stp = one
|
|
endif
|
|
|
|
call dcopy(n,x,1,t,1)
|
|
call dcopy(n,g,1,r,1)
|
|
fold = f
|
|
ifun = 0
|
|
iback = 0
|
|
csave = 'START'
|
|
556 continue
|
|
gd = ddot(n,g,1,d,1)
|
|
if (ifun .eq. 0) then
|
|
gdold=gd
|
|
if (gd .ge. zero) then
|
|
c the directional derivative >=0.
|
|
c Line search is impossible.
|
|
write(6,*)' ascent direction in projection gd = ', gd
|
|
info = -4
|
|
return
|
|
endif
|
|
endif
|
|
|
|
call dcsrch(f,gd,stp,ftol,gtol,xtol,zero,stpmx,csave,isave,dsave)
|
|
|
|
xstep = stp*dnorm
|
|
if (csave(1:4) .ne. 'CONV' .and. csave(1:4) .ne. 'WARN') then
|
|
task = 'FG_LNSRCH'
|
|
ifun = ifun + 1
|
|
nfgv = nfgv + 1
|
|
iback = ifun - 1
|
|
if (stp .eq. one) then
|
|
call dcopy(n,z,1,x,1)
|
|
else
|
|
do 41 i = 1, n
|
|
x(i) = stp*d(i) + t(i)
|
|
41 continue
|
|
endif
|
|
else
|
|
task = 'NEW_X'
|
|
endif
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of lnsrlb =============================
|
|
|
|
subroutine matupd(n, m, ws, wy, sy, ss, d, r, itail,
|
|
+ iupdat, col, head, theta, rr, dr, stp, dtd)
|
|
|
|
integer n, m, itail, iupdat, col, head
|
|
double precision theta, rr, dr, stp, dtd, d(n), r(n),
|
|
+ ws(n, m), wy(n, m), sy(m, m), ss(m, m)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine matupd
|
|
c
|
|
c This subroutine updates matrices WS and WY, and forms the
|
|
c middle matrix in B.
|
|
c
|
|
c Subprograms called:
|
|
c
|
|
c Linpack ... dcopy, ddot.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer j,pointr
|
|
double precision ddot
|
|
double precision one
|
|
parameter (one=1.0d0)
|
|
|
|
c Set pointers for matrices WS and WY.
|
|
|
|
if (iupdat .le. m) then
|
|
col = iupdat
|
|
itail = mod(head+iupdat-2,m) + 1
|
|
else
|
|
itail = mod(itail,m) + 1
|
|
head = mod(head,m) + 1
|
|
endif
|
|
|
|
c Update matrices WS and WY.
|
|
|
|
call dcopy(n,d,1,ws(1,itail),1)
|
|
call dcopy(n,r,1,wy(1,itail),1)
|
|
|
|
c Set theta=yy/ys.
|
|
|
|
theta = rr/dr
|
|
|
|
c Form the middle matrix in B.
|
|
|
|
c update the upper triangle of SS,
|
|
c and the lower triangle of SY:
|
|
if (iupdat .gt. m) then
|
|
c move old information
|
|
do 50 j = 1, col - 1
|
|
call dcopy(j,ss(2,j+1),1,ss(1,j),1)
|
|
call dcopy(col-j,sy(j+1,j+1),1,sy(j,j),1)
|
|
50 continue
|
|
endif
|
|
c add new information: the last row of SY
|
|
c and the last column of SS:
|
|
pointr = head
|
|
do 51 j = 1, col - 1
|
|
sy(col,j) = ddot(n,d,1,wy(1,pointr),1)
|
|
ss(j,col) = ddot(n,ws(1,pointr),1,d,1)
|
|
pointr = mod(pointr,m) + 1
|
|
51 continue
|
|
if (stp .eq. one) then
|
|
ss(col,col) = dtd
|
|
else
|
|
ss(col,col) = stp*stp*dtd
|
|
endif
|
|
sy(col,col) = dr
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of matupd =============================
|
|
|
|
subroutine prn1lb(n, m, l, u, x, iprint, itfile, epsmch)
|
|
|
|
integer n, m, iprint, itfile
|
|
double precision epsmch, x(n), l(n), u(n)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine prn1lb
|
|
c
|
|
c This subroutine prints the input data, initial point, upper and
|
|
c lower bounds of each variable, machine precision, as well as
|
|
c the headings of the output.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer i
|
|
|
|
if (iprint .ge. 0) then
|
|
write (6,7001) epsmch
|
|
write (6,*) 'N = ',n,' M = ',m
|
|
if (iprint .ge. 1) then
|
|
write (itfile,2001) epsmch
|
|
write (itfile,*)'N = ',n,' M = ',m
|
|
write (itfile,9001)
|
|
if (iprint .gt. 100) then
|
|
write (6,1004) 'L =',(l(i),i = 1,n)
|
|
write (6,1004) 'X0 =',(x(i),i = 1,n)
|
|
write (6,1004) 'U =',(u(i),i = 1,n)
|
|
endif
|
|
endif
|
|
endif
|
|
|
|
1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4)))
|
|
2001 format ('RUNNING THE L-BFGS-B CODE',/,/,
|
|
+ 'it = iteration number',/,
|
|
+ 'nf = number of function evaluations',/,
|
|
+ 'nseg = number of segments explored during the Cauchy search',/,
|
|
+ 'nact = number of active bounds at the generalized Cauchy point'
|
|
+ ,/,
|
|
+ 'sub = manner in which the subspace minimization terminated:'
|
|
+ ,/,' con = converged, bnd = a bound was reached',/,
|
|
+ 'itls = number of iterations performed in the line search',/,
|
|
+ 'stepl = step length used',/,
|
|
+ 'tstep = norm of the displacement (total step)',/,
|
|
+ 'projg = norm of the projected gradient',/,
|
|
+ 'f = function value',/,/,
|
|
+ ' * * *',/,/,
|
|
+ 'Machine precision =',1p,d10.3)
|
|
7001 format ('RUNNING THE L-BFGS-B CODE',/,/,
|
|
+ ' * * *',/,/,
|
|
+ 'Machine precision =',1p,d10.3)
|
|
9001 format (/,3x,'it',3x,'nf',2x,'nseg',2x,'nact',2x,'sub',2x,'itls',
|
|
+ 2x,'stepl',4x,'tstep',5x,'projg',8x,'f')
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of prn1lb =============================
|
|
|
|
subroutine prn2lb(n, x, f, g, iprint, itfile, iter, nfgv, nact,
|
|
+ sbgnrm, nseg, word, iword, iback, stp, xstep)
|
|
|
|
!(ajt 8/9/13)
|
|
USE LOGICAL_ADJ_MOD, ONLY: LATF
|
|
|
|
character*3 word
|
|
integer n, iprint, itfile, iter, nfgv, nact, nseg,
|
|
+ iword, iback
|
|
double precision f, sbgnrm, stp, xstep, x(n), g(n)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine prn2lb
|
|
c
|
|
c This subroutine prints out new information after a successful
|
|
c line search.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer i,imod
|
|
|
|
! (ajt, 8/9/13)
|
|
LATF = .TRUE.
|
|
|
|
c 'word' records the status of subspace solutions.
|
|
if (iword .eq. 0) then
|
|
c the subspace minimization converged.
|
|
word = 'con'
|
|
else if (iword .eq. 1) then
|
|
c the subspace minimization stopped at a bound.
|
|
word = 'bnd'
|
|
else if (iword .eq. 5) then
|
|
c the truncated Newton step has been used.
|
|
word = 'TNT'
|
|
else
|
|
word = '---'
|
|
endif
|
|
if (iprint .ge. 99) then
|
|
write (6,*) 'LINE SEARCH',iback,' times; norm of step = ',xstep
|
|
write (6,2001) iter,f,sbgnrm
|
|
if (iprint .gt. 100) then
|
|
write (6,1004) 'X =',(x(i), i = 1, n)
|
|
write (6,1004) 'G =',(g(i), i = 1, n)
|
|
endif
|
|
else if (iprint .gt. 0) then
|
|
imod = mod(iter,iprint)
|
|
if (imod .eq. 0) write (6,2001) iter,f,sbgnrm
|
|
endif
|
|
if (iprint .ge. 1) write (itfile,3001)
|
|
+ iter,nfgv,nseg,nact,word,iback,stp,xstep,sbgnrm,f
|
|
|
|
1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4)))
|
|
2001 format
|
|
+ (/,'At iterate',i5,4x,'f= ',1p,d12.5,4x,'|proj g|= ',1p,d12.5)
|
|
3001 format(2(1x,i4),2(1x,i5),2x,a3,1x,i4,1p,2(2x,d7.1),1p,2(1x,d10.3))
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of prn2lb =============================
|
|
|
|
subroutine prn3lb(n, x, f, task, iprint, info, itfile,
|
|
+ iter, nfgv, nintol, nskip, nact, sbgnrm,
|
|
+ time, nseg, word, iback, stp, xstep, k,
|
|
+ cachyt, sbtime, lnscht)
|
|
|
|
character*60 task
|
|
character*3 word
|
|
integer n, iprint, info, itfile, iter, nfgv, nintol,
|
|
+ nskip, nact, nseg, iback, k
|
|
double precision f, sbgnrm, time, stp, xstep, cachyt, sbtime,
|
|
+ lnscht, x(n)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine prn3lb
|
|
c
|
|
c This subroutine prints out information when either a built-in
|
|
c convergence test is satisfied or when an error message is
|
|
c generated.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer i
|
|
|
|
if (task(1:5) .eq. 'ERROR') goto 999
|
|
|
|
if (iprint .ge. 0) then
|
|
write (6,3003)
|
|
write (6,3004)
|
|
write(6,3005) n,iter,nfgv,nintol,nskip,nact,sbgnrm,f
|
|
if (iprint .ge. 100) then
|
|
write (6,1004) 'X =',(x(i),i = 1,n)
|
|
endif
|
|
if (iprint .ge. 1) write (6,*) ' F =',f
|
|
endif
|
|
999 continue
|
|
if (iprint .ge. 0) then
|
|
write (6,3009) task
|
|
if (info .ne. 0) then
|
|
if (info .eq. -1) write (6,9011)
|
|
if (info .eq. -2) write (6,9012)
|
|
if (info .eq. -3) write (6,9013)
|
|
if (info .eq. -4) write (6,9014)
|
|
if (info .eq. -5) write (6,9015)
|
|
if (info .eq. -6) write (6,*)' Input nbd(',k,') is invalid.'
|
|
if (info .eq. -7)
|
|
+ write (6,*)' l(',k,') > u(',k,'). No feasible solution.'
|
|
if (info .eq. -8) write (6,9018)
|
|
if (info .eq. -9) write (6,9019)
|
|
endif
|
|
if (iprint .ge. 1) write (6,3007) cachyt,sbtime,lnscht
|
|
write (6,3008) time
|
|
if (iprint .ge. 1) then
|
|
if (info .eq. -4 .or. info .eq. -9) then
|
|
write (itfile,3002)
|
|
+ iter,nfgv,nseg,nact,word,iback,stp,xstep
|
|
endif
|
|
write (itfile,3009) task
|
|
if (info .ne. 0) then
|
|
if (info .eq. -1) write (itfile,9011)
|
|
if (info .eq. -2) write (itfile,9012)
|
|
if (info .eq. -3) write (itfile,9013)
|
|
if (info .eq. -4) write (itfile,9014)
|
|
if (info .eq. -5) write (itfile,9015)
|
|
if (info .eq. -8) write (itfile,9018)
|
|
if (info .eq. -9) write (itfile,9019)
|
|
endif
|
|
write (itfile,3008) time
|
|
endif
|
|
endif
|
|
|
|
1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4)))
|
|
3002 format(2(1x,i4),2(1x,i5),2x,a3,1x,i4,1p,2(2x,d7.1),6x,'-',10x,'-')
|
|
3003 format (/,
|
|
+ ' * * *',/,/,
|
|
+ 'Tit = total number of iterations',/,
|
|
+ 'Tnf = total number of function evaluations',/,
|
|
+ 'Tnint = total number of segments explored during',
|
|
+ ' Cauchy searches',/,
|
|
+ 'Skip = number of BFGS updates skipped',/,
|
|
+ 'Nact = number of active bounds at final generalized',
|
|
+ ' Cauchy point',/,
|
|
+ 'Projg = norm of the final projected gradient',/,
|
|
+ 'F = final function value',/,/,
|
|
+ ' * * *')
|
|
3004 format (/,3x,'N',4x,'Tit',5x,'Tnf',2x,'Tnint',2x,
|
|
+ 'Skip',2x,'Nact',5x,'Projg',8x,'F')
|
|
3005 format (i5,2(1x,i6),(1x,i6),(2x,i4),(1x,i5),1p,2(2x,d10.3))
|
|
3007 format (/,' Cauchy time',1p,e10.3,' seconds.',/
|
|
+ ' Subspace minimization time',1p,e10.3,' seconds.',/
|
|
+ ' Line search time',1p,e10.3,' seconds.')
|
|
3008 format (/,' Total User time',1p,e10.3,' seconds.',/)
|
|
3009 format (/,a60)
|
|
9011 format (/,
|
|
+' Matrix in 1st Cholesky factorization in formk is not Pos. Def.')
|
|
9012 format (/,
|
|
+' Matrix in 2st Cholesky factorization in formk is not Pos. Def.')
|
|
9013 format (/,
|
|
+' Matrix in the Cholesky factorization in formt is not Pos. Def.')
|
|
9014 format (/,
|
|
+' Derivative >= 0, backtracking line search impossible.',/,
|
|
+' Previous x, f and g restored.',/,
|
|
+' Possible causes: 1 error in function or gradient evaluation;',/,
|
|
+' 2 rounding errors dominate computation.')
|
|
9015 format (/,
|
|
+' Warning: more than 10 function and gradient',/,
|
|
+' evaluations in the last line search. Termination',/,
|
|
+' may possibly be caused by a bad search direction.')
|
|
9018 format (/,' The triangular system is singular.')
|
|
9019 format (/,
|
|
+' Line search cannot locate an adequate point after 20 function',/
|
|
+,' and gradient evaluations. Previous x, f and g restored.',/,
|
|
+' Possible causes: 1 error in function or gradient evaluation;',/,
|
|
+' 2 rounding error dominate computation.')
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of prn3lb =============================
|
|
|
|
subroutine projgr(n, l, u, nbd, x, g, sbgnrm)
|
|
|
|
integer n, nbd(n)
|
|
double precision sbgnrm, x(n), l(n), u(n), g(n)
|
|
|
|
c ************
|
|
c
|
|
c Subroutine projgr
|
|
c
|
|
c This subroutine computes the infinity norm of the projected
|
|
c gradient.
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer i
|
|
double precision gi
|
|
double precision one,zero
|
|
parameter (one=1.0d0,zero=0.0d0)
|
|
|
|
sbgnrm = zero
|
|
do 15 i = 1, n
|
|
gi = g(i)
|
|
if (nbd(i) .ne. 0) then
|
|
if (gi .lt. zero) then
|
|
if (nbd(i) .ge. 2) gi = max((x(i)-u(i)),gi)
|
|
else
|
|
if (nbd(i) .le. 2) gi = min((x(i)-l(i)),gi)
|
|
endif
|
|
endif
|
|
sbgnrm = max(sbgnrm,abs(gi))
|
|
15 continue
|
|
|
|
return
|
|
|
|
end
|
|
|
|
c======================= The end of projgr =============================
|
|
|
|
subroutine subsm ( n, m, nsub, ind, l, u, nbd, x, d, xp, ws, wy,
|
|
+ theta, xx, gg,
|
|
+ col, head, iword, wv, wn, iprint, info )
|
|
implicit none
|
|
integer n, m, nsub, col, head, iword, iprint, info,
|
|
+ ind(nsub), nbd(n)
|
|
double precision theta,
|
|
+ l(n), u(n), x(n), d(n), xp(n), xx(n), gg(n),
|
|
+ ws(n, m), wy(n, m),
|
|
+ wv(2*m), wn(2*m, 2*m)
|
|
|
|
c **********************************************************************
|
|
c
|
|
c This routine contains the major changes in the updated version.
|
|
c The changes are described in the accompanying paper
|
|
c
|
|
c Jose Luis Morales, Jorge Nocedal
|
|
c "Remark On Algorithm 788: L-BFGS-B: Fortran Subroutines for Large-Scale
|
|
c Bound Constrained Optimization". Decemmber 27, 2010.
|
|
c
|
|
c J.L. Morales Departamento de Matematicas,
|
|
c Instituto Tecnologico Autonomo de Mexico
|
|
c Mexico D.F.
|
|
c
|
|
c J, Nocedal Department of Electrical Engineering and
|
|
c Computer Science.
|
|
c Northwestern University. Evanston, IL. USA
|
|
c
|
|
c January 17, 2011
|
|
c
|
|
c **********************************************************************
|
|
c
|
|
c
|
|
c Subroutine subsm
|
|
c
|
|
c Given xcp, l, u, r, an index set that specifies
|
|
c the active set at xcp, and an l-BFGS matrix B
|
|
c (in terms of WY, WS, SY, WT, head, col, and theta),
|
|
c this subroutine computes an approximate solution
|
|
c of the subspace problem
|
|
c
|
|
c (P) min Q(x) = r'(x-xcp) + 1/2 (x-xcp)' B (x-xcp)
|
|
c
|
|
c subject to l<=x<=u
|
|
c x_i=xcp_i for all i in A(xcp)
|
|
c
|
|
c along the subspace unconstrained Newton direction
|
|
c
|
|
c d = -(Z'BZ)^(-1) r.
|
|
c
|
|
c The formula for the Newton direction, given the L-BFGS matrix
|
|
c and the Sherman-Morrison formula, is
|
|
c
|
|
c d = (1/theta)r + (1/theta*2) Z'WK^(-1)W'Z r.
|
|
c
|
|
c where
|
|
c K = [-D -Y'ZZ'Y/theta L_a'-R_z' ]
|
|
c [L_a -R_z theta*S'AA'S ]
|
|
c
|
|
c Note that this procedure for computing d differs
|
|
c from that described in [1]. One can show that the matrix K is
|
|
c equal to the matrix M^[-1]N in that paper.
|
|
c
|
|
c n is an integer variable.
|
|
c On entry n is the dimension of the problem.
|
|
c On exit n is unchanged.
|
|
c
|
|
c m is an integer variable.
|
|
c On entry m is the maximum number of variable metric corrections
|
|
c used to define the limited memory matrix.
|
|
c On exit m is unchanged.
|
|
c
|
|
c nsub is an integer variable.
|
|
c On entry nsub is the number of free variables.
|
|
c On exit nsub is unchanged.
|
|
c
|
|
c ind is an integer array of dimension nsub.
|
|
c On entry ind specifies the coordinate indices of free variables.
|
|
c On exit ind is unchanged.
|
|
c
|
|
c l is a double precision array of dimension n.
|
|
c On entry l is the lower bound of x.
|
|
c On exit l is unchanged.
|
|
c
|
|
c u is a double precision array of dimension n.
|
|
c On entry u is the upper bound of x.
|
|
c On exit u is unchanged.
|
|
c
|
|
c nbd is a integer array of dimension n.
|
|
c On entry nbd represents the type of bounds imposed on the
|
|
c variables, and must be specified as follows:
|
|
c nbd(i)=0 if x(i) is unbounded,
|
|
c 1 if x(i) has only a lower bound,
|
|
c 2 if x(i) has both lower and upper bounds, and
|
|
c 3 if x(i) has only an upper bound.
|
|
c On exit nbd is unchanged.
|
|
c
|
|
c x is a double precision array of dimension n.
|
|
c On entry x specifies the Cauchy point xcp.
|
|
c On exit x(i) is the minimizer of Q over the subspace of
|
|
c free variables.
|
|
c
|
|
c d is a double precision array of dimension n.
|
|
c On entry d is the reduced gradient of Q at xcp.
|
|
c On exit d is the Newton direction of Q.
|
|
c
|
|
c xp is a double precision array of dimension n.
|
|
c used to safeguard the projected Newton direction
|
|
c
|
|
c xx is a double precision array of dimension n
|
|
c On entry it holds the current iterate
|
|
c On output it is unchanged
|
|
|
|
c gg is a double precision array of dimension n
|
|
c On entry it holds the gradient at the current iterate
|
|
c On output it is unchanged
|
|
c
|
|
c ws and wy are double precision arrays;
|
|
c theta is a double precision variable;
|
|
c col is an integer variable;
|
|
c head is an integer variable.
|
|
c On entry they store the information defining the
|
|
c limited memory BFGS matrix:
|
|
c ws(n,m) stores S, a set of s-vectors;
|
|
c wy(n,m) stores Y, a set of y-vectors;
|
|
c theta is the scaling factor specifying B_0 = theta I;
|
|
c col is the number of variable metric corrections stored;
|
|
c head is the location of the 1st s- (or y-) vector in S (or Y).
|
|
c On exit they are unchanged.
|
|
c
|
|
c iword is an integer variable.
|
|
c On entry iword is unspecified.
|
|
c On exit iword specifies the status of the subspace solution.
|
|
c iword = 0 if the solution is in the box,
|
|
c 1 if some bound is encountered.
|
|
c
|
|
c wv is a double precision working array of dimension 2m.
|
|
c
|
|
c wn is a double precision array of dimension 2m x 2m.
|
|
c On entry the upper triangle of wn stores the LEL^T factorization
|
|
c of the indefinite matrix
|
|
c
|
|
c K = [-D -Y'ZZ'Y/theta L_a'-R_z' ]
|
|
c [L_a -R_z theta*S'AA'S ]
|
|
c where E = [-I 0]
|
|
c [ 0 I]
|
|
c On exit wn is unchanged.
|
|
c
|
|
c iprint is an INTEGER variable that must be set by the user.
|
|
c It controls the frequency and type of output generated:
|
|
c iprint<0 no output is generated;
|
|
c iprint=0 print only one line at the last iteration;
|
|
c 0<iprint<99 print also f and |proj g| every iprint iterations;
|
|
c iprint=99 print details of every iteration except n-vectors;
|
|
c iprint=100 print also the changes of active set and final x;
|
|
c iprint>100 print details of every iteration including x and g;
|
|
c When iprint > 0, the file iterate.dat will be created to
|
|
c summarize the iteration.
|
|
c
|
|
c info is an integer variable.
|
|
c On entry info is unspecified.
|
|
c On exit info = 0 for normal return,
|
|
c = nonzero for abnormal return
|
|
c when the matrix K is ill-conditioned.
|
|
c
|
|
c Subprograms called:
|
|
c
|
|
c Linpack dtrsl.
|
|
c
|
|
c
|
|
c References:
|
|
c
|
|
c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
|
|
c memory algorithm for bound constrained optimization'',
|
|
c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.
|
|
c
|
|
c
|
|
c
|
|
c * * *
|
|
c
|
|
c NEOS, November 1994. (Latest revision June 1996.)
|
|
c Optimization Technology Center.
|
|
c Argonne National Laboratory and Northwestern University.
|
|
c Written by
|
|
c Ciyou Zhu
|
|
c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
|
|
c
|
|
c
|
|
c ************
|
|
|
|
integer pointr,m2,col2,ibd,jy,js,i,j,k
|
|
double precision alpha, xk, dk, temp1, temp2
|
|
double precision one,zero
|
|
parameter (one=1.0d0,zero=0.0d0)
|
|
c
|
|
double precision dd_p
|
|
|
|
if (nsub .le. 0) return
|
|
if (iprint .ge. 99) write (6,1001)
|
|
|
|
c Compute wv = W'Zd.
|
|
|
|
pointr = head
|
|
do 20 i = 1, col
|
|
temp1 = zero
|
|
temp2 = zero
|
|
do 10 j = 1, nsub
|
|
k = ind(j)
|
|
temp1 = temp1 + wy(k,pointr)*d(j)
|
|
temp2 = temp2 + ws(k,pointr)*d(j)
|
|
10 continue
|
|
wv(i) = temp1
|
|
wv(col + i) = theta*temp2
|
|
pointr = mod(pointr,m) + 1
|
|
20 continue
|
|
|
|
c Compute wv:=K^(-1)wv.
|
|
|
|
m2 = 2*m
|
|
col2 = 2*col
|
|
call dtrsl(wn,m2,col2,wv,11,info)
|
|
if (info .ne. 0) return
|
|
do 25 i = 1, col
|
|
wv(i) = -wv(i)
|
|
25 continue
|
|
call dtrsl(wn,m2,col2,wv,01,info)
|
|
if (info .ne. 0) return
|
|
|
|
c Compute d = (1/theta)d + (1/theta**2)Z'W wv.
|
|
|
|
pointr = head
|
|
do 40 jy = 1, col
|
|
js = col + jy
|
|
do 30 i = 1, nsub
|
|
k = ind(i)
|
|
d(i) = d(i) + wy(k,pointr)*wv(jy)/theta
|
|
+ + ws(k,pointr)*wv(js)
|
|
30 continue
|
|
pointr = mod(pointr,m) + 1
|
|
40 continue
|
|
|
|
call dscal( nsub, one/theta, d, 1 )
|
|
c
|
|
c-----------------------------------------------------------------
|
|
c Let us try the projection, d is the Newton direction
|
|
|
|
iword = 0
|
|
|
|
call dcopy ( n, x, 1, xp, 1 )
|
|
c
|
|
do 50 i=1, nsub
|
|
k = ind(i)
|
|
dk = d(i)
|
|
xk = x(k)
|
|
if ( nbd(k) .ne. 0 ) then
|
|
c
|
|
if ( nbd(k).eq.1 ) then ! lower bounds only
|
|
x(k) = max( l(k), xk + dk )
|
|
if ( x(k).eq.l(k) ) iword = 1
|
|
else
|
|
c
|
|
if ( nbd(k).eq.2 ) then ! upper and lower bounds
|
|
xk = max( l(k), xk + dk )
|
|
x(k) = min( u(k), xk )
|
|
if ( x(k).eq.l(k) .or. x(k).eq.u(k) ) iword = 1
|
|
else
|
|
c
|
|
if ( nbd(k).eq.3 ) then ! upper bounds only
|
|
x(k) = min( u(k), xk + dk )
|
|
if ( x(k).eq.u(k) ) iword = 1
|
|
end if
|
|
end if
|
|
end if
|
|
c
|
|
else ! free variables
|
|
x(k) = xk + dk
|
|
end if
|
|
50 continue
|
|
c
|
|
if ( iword.eq.0 ) then
|
|
go to 911
|
|
end if
|
|
c
|
|
c check sign of the directional derivative
|
|
c
|
|
dd_p = zero
|
|
do 55 i=1, n
|
|
dd_p = dd_p + (x(i) - xx(i))*gg(i)
|
|
55 continue
|
|
if ( dd_p .gt.zero ) then
|
|
call dcopy( n, xp, 1, x, 1 )
|
|
write(6,*) ' Positive dir derivative in projection '
|
|
write(6,*) ' Using the backtracking step '
|
|
else
|
|
go to 911
|
|
endif
|
|
c
|
|
c-----------------------------------------------------------------
|
|
c
|
|
alpha = one
|
|
temp1 = alpha
|
|
ibd = 0
|
|
do 60 i = 1, nsub
|
|
k = ind(i)
|
|
dk = d(i)
|
|
if (nbd(k) .ne. 0) then
|
|
if (dk .lt. zero .and. nbd(k) .le. 2) then
|
|
temp2 = l(k) - x(k)
|
|
if (temp2 .ge. zero) then
|
|
temp1 = zero
|
|
else if (dk*alpha .lt. temp2) then
|
|
temp1 = temp2/dk
|
|
endif
|
|
else if (dk .gt. zero .and. nbd(k) .ge. 2) then
|
|
temp2 = u(k) - x(k)
|
|
if (temp2 .le. zero) then
|
|
temp1 = zero
|
|
else if (dk*alpha .gt. temp2) then
|
|
temp1 = temp2/dk
|
|
endif
|
|
endif
|
|
if (temp1 .lt. alpha) then
|
|
alpha = temp1
|
|
ibd = i
|
|
endif
|
|
endif
|
|
60 continue
|
|
|
|
if (alpha .lt. one) then
|
|
dk = d(ibd)
|
|
k = ind(ibd)
|
|
if (dk .gt. zero) then
|
|
x(k) = u(k)
|
|
d(ibd) = zero
|
|
else if (dk .lt. zero) then
|
|
x(k) = l(k)
|
|
d(ibd) = zero
|
|
endif
|
|
endif
|
|
do 70 i = 1, nsub
|
|
k = ind(i)
|
|
x(k) = x(k) + alpha*d(i)
|
|
70 continue
|
|
cccccc
|
|
911 continue
|
|
|
|
if (iprint .ge. 99) write (6,1004)
|
|
|
|
1001 format (/,'----------------SUBSM entered-----------------',/)
|
|
1004 format (/,'----------------exit SUBSM --------------------',/)
|
|
|
|
return
|
|
|
|
end
|
|
c====================== The end of subsm ===============================
|
|
|
|
subroutine dcsrch(f,g,stp,ftol,gtol,xtol,stpmin,stpmax,
|
|
+ task,isave,dsave)
|
|
character*(*) task
|
|
integer isave(2)
|
|
double precision f,g,stp,ftol,gtol,xtol,stpmin,stpmax
|
|
double precision dsave(13)
|
|
c **********
|
|
c
|
|
c Subroutine dcsrch
|
|
c
|
|
c This subroutine finds a step that satisfies a sufficient
|
|
c decrease condition and a curvature condition.
|
|
c
|
|
c Each call of the subroutine updates an interval with
|
|
c endpoints stx and sty. The interval is initially chosen
|
|
c so that it contains a minimizer of the modified function
|
|
c
|
|
c psi(stp) = f(stp) - f(0) - ftol*stp*f'(0).
|
|
c
|
|
c If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the
|
|
c interval is chosen so that it contains a minimizer of f.
|
|
c
|
|
c The algorithm is designed to find a step that satisfies
|
|
c the sufficient decrease condition
|
|
c
|
|
c f(stp) <= f(0) + ftol*stp*f'(0),
|
|
c
|
|
c and the curvature condition
|
|
c
|
|
c abs(f'(stp)) <= gtol*abs(f'(0)).
|
|
c
|
|
c If ftol is less than gtol and if, for example, the function
|
|
c is bounded below, then there is always a step which satisfies
|
|
c both conditions.
|
|
c
|
|
c If no step can be found that satisfies both conditions, then
|
|
c the algorithm stops with a warning. In this case stp only
|
|
c satisfies the sufficient decrease condition.
|
|
c
|
|
c A typical invocation of dcsrch has the following outline:
|
|
c
|
|
c task = 'START'
|
|
c 10 continue
|
|
c call dcsrch( ... )
|
|
c if (task .eq. 'FG') then
|
|
c Evaluate the function and the gradient at stp
|
|
c goto 10
|
|
c end if
|
|
c
|
|
c NOTE: The user must no alter work arrays between calls.
|
|
c
|
|
c The subroutine statement is
|
|
c
|
|
c subroutine dcsrch(f,g,stp,ftol,gtol,xtol,stpmin,stpmax,
|
|
c task,isave,dsave)
|
|
c where
|
|
c
|
|
c f is a double precision variable.
|
|
c On initial entry f is the value of the function at 0.
|
|
c On subsequent entries f is the value of the
|
|
c function at stp.
|
|
c On exit f is the value of the function at stp.
|
|
c
|
|
c g is a double precision variable.
|
|
c On initial entry g is the derivative of the function at 0.
|
|
c On subsequent entries g is the derivative of the
|
|
c function at stp.
|
|
c On exit g is the derivative of the function at stp.
|
|
c
|
|
c stp is a double precision variable.
|
|
c On entry stp is the current estimate of a satisfactory
|
|
c step. On initial entry, a positive initial estimate
|
|
c must be provided.
|
|
c On exit stp is the current estimate of a satisfactory step
|
|
c if task = 'FG'. If task = 'CONV' then stp satisfies
|
|
c the sufficient decrease and curvature condition.
|
|
c
|
|
c ftol is a double precision variable.
|
|
c On entry ftol specifies a nonnegative tolerance for the
|
|
c sufficient decrease condition.
|
|
c On exit ftol is unchanged.
|
|
c
|
|
c gtol is a double precision variable.
|
|
c On entry gtol specifies a nonnegative tolerance for the
|
|
c curvature condition.
|
|
c On exit gtol is unchanged.
|
|
c
|
|
c xtol is a double precision variable.
|
|
c On entry xtol specifies a nonnegative relative tolerance
|
|
c for an acceptable step. The subroutine exits with a
|
|
c warning if the relative difference between sty and stx
|
|
c is less than xtol.
|
|
c On exit xtol is unchanged.
|
|
c
|
|
c stpmin is a double precision variable.
|
|
c On entry stpmin is a nonnegative lower bound for the step.
|
|
c On exit stpmin is unchanged.
|
|
c
|
|
c stpmax is a double precision variable.
|
|
c On entry stpmax is a nonnegative upper bound for the step.
|
|
c On exit stpmax is unchanged.
|
|
c
|
|
c task is a character variable of length at least 60.
|
|
c On initial entry task must be set to 'START'.
|
|
c On exit task indicates the required action:
|
|
c
|
|
c If task(1:2) = 'FG' then evaluate the function and
|
|
c derivative at stp and call dcsrch again.
|
|
c
|
|
c If task(1:4) = 'CONV' then the search is successful.
|
|
c
|
|
c If task(1:4) = 'WARN' then the subroutine is not able
|
|
c to satisfy the convergence conditions. The exit value of
|
|
c stp contains the best point found during the search.
|
|
c
|
|
c If task(1:5) = 'ERROR' then there is an error in the
|
|
c input arguments.
|
|
c
|
|
c On exit with convergence, a warning or an error, the
|
|
c variable task contains additional information.
|
|
c
|
|
c isave is an integer work array of dimension 2.
|
|
c
|
|
c dsave is a double precision work array of dimension 13.
|
|
c
|
|
c Subprograms called
|
|
c
|
|
c MINPACK-2 ... dcstep
|
|
c
|
|
c MINPACK-1 Project. June 1983.
|
|
c Argonne National Laboratory.
|
|
c Jorge J. More' and David J. Thuente.
|
|
c
|
|
c MINPACK-2 Project. October 1993.
|
|
c Argonne National Laboratory and University of Minnesota.
|
|
c Brett M. Averick, Richard G. Carter, and Jorge J. More'.
|
|
c
|
|
c **********
|
|
double precision zero,p5,p66
|
|
parameter(zero=0.0d0,p5=0.5d0,p66=0.66d0)
|
|
double precision xtrapl,xtrapu
|
|
parameter(xtrapl=1.1d0,xtrapu=4.0d0)
|
|
|
|
logical brackt
|
|
integer stage
|
|
double precision finit,ftest,fm,fx,fxm,fy,fym,ginit,gtest,
|
|
+ gm,gx,gxm,gy,gym,stx,sty,stmin,stmax,width,width1
|
|
|
|
c Initialization block.
|
|
|
|
if (task(1:5) .eq. 'START') then
|
|
|
|
c Check the input arguments for errors.
|
|
|
|
if (stp .lt. stpmin) task = 'ERROR: STP .LT. STPMIN'
|
|
if (stp .gt. stpmax) task = 'ERROR: STP .GT. STPMAX'
|
|
if (g .ge. zero) task = 'ERROR: INITIAL G .GE. ZERO'
|
|
if (ftol .lt. zero) task = 'ERROR: FTOL .LT. ZERO'
|
|
if (gtol .lt. zero) task = 'ERROR: GTOL .LT. ZERO'
|
|
if (xtol .lt. zero) task = 'ERROR: XTOL .LT. ZERO'
|
|
if (stpmin .lt. zero) task = 'ERROR: STPMIN .LT. ZERO'
|
|
if (stpmax .lt. stpmin) task = 'ERROR: STPMAX .LT. STPMIN'
|
|
|
|
c Exit if there are errors on input.
|
|
|
|
if (task(1:5) .eq. 'ERROR') return
|
|
|
|
c Initialize local variables.
|
|
|
|
brackt = .false.
|
|
stage = 1
|
|
finit = f
|
|
ginit = g
|
|
gtest = ftol*ginit
|
|
width = stpmax - stpmin
|
|
width1 = width/p5
|
|
|
|
c The variables stx, fx, gx contain the values of the step,
|
|
c function, and derivative at the best step.
|
|
c The variables sty, fy, gy contain the value of the step,
|
|
c function, and derivative at sty.
|
|
c The variables stp, f, g contain the values of the step,
|
|
c function, and derivative at stp.
|
|
|
|
stx = zero
|
|
fx = finit
|
|
gx = ginit
|
|
sty = zero
|
|
fy = finit
|
|
gy = ginit
|
|
stmin = zero
|
|
stmax = stp + xtrapu*stp
|
|
task = 'FG'
|
|
|
|
goto 1000
|
|
|
|
else
|
|
|
|
c Restore local variables.
|
|
|
|
if (isave(1) .eq. 1) then
|
|
brackt = .true.
|
|
else
|
|
brackt = .false.
|
|
endif
|
|
stage = isave(2)
|
|
ginit = dsave(1)
|
|
gtest = dsave(2)
|
|
gx = dsave(3)
|
|
gy = dsave(4)
|
|
finit = dsave(5)
|
|
fx = dsave(6)
|
|
fy = dsave(7)
|
|
stx = dsave(8)
|
|
sty = dsave(9)
|
|
stmin = dsave(10)
|
|
stmax = dsave(11)
|
|
width = dsave(12)
|
|
width1 = dsave(13)
|
|
|
|
endif
|
|
|
|
c If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the
|
|
c algorithm enters the second stage.
|
|
|
|
ftest = finit + stp*gtest
|
|
if (stage .eq. 1 .and. f .le. ftest .and. g .ge. zero)
|
|
+ stage = 2
|
|
|
|
c Test for warnings.
|
|
|
|
if (brackt .and. (stp .le. stmin .or. stp .ge. stmax))
|
|
+ task = 'WARNING: ROUNDING ERRORS PREVENT PROGRESS'
|
|
if (brackt .and. stmax - stmin .le. xtol*stmax)
|
|
+ task = 'WARNING: XTOL TEST SATISFIED'
|
|
if (stp .eq. stpmax .and. f .le. ftest .and. g .le. gtest)
|
|
+ task = 'WARNING: STP = STPMAX'
|
|
if (stp .eq. stpmin .and. (f .gt. ftest .or. g .ge. gtest))
|
|
+ task = 'WARNING: STP = STPMIN'
|
|
|
|
c Test for convergence.
|
|
|
|
if (f .le. ftest .and. abs(g) .le. gtol*(-ginit))
|
|
+ task = 'CONVERGENCE'
|
|
|
|
c Test for termination.
|
|
|
|
if (task(1:4) .eq. 'WARN' .or. task(1:4) .eq. 'CONV') goto 1000
|
|
|
|
c A modified function is used to predict the step during the
|
|
c first stage if a lower function value has been obtained but
|
|
c the decrease is not sufficient.
|
|
|
|
if (stage .eq. 1 .and. f .le. fx .and. f .gt. ftest) then
|
|
|
|
c Define the modified function and derivative values.
|
|
|
|
fm = f - stp*gtest
|
|
fxm = fx - stx*gtest
|
|
fym = fy - sty*gtest
|
|
gm = g - gtest
|
|
gxm = gx - gtest
|
|
gym = gy - gtest
|
|
|
|
c Call dcstep to update stx, sty, and to compute the new step.
|
|
|
|
call dcstep(stx,fxm,gxm,sty,fym,gym,stp,fm,gm,
|
|
+ brackt,stmin,stmax)
|
|
|
|
c Reset the function and derivative values for f.
|
|
|
|
fx = fxm + stx*gtest
|
|
fy = fym + sty*gtest
|
|
gx = gxm + gtest
|
|
gy = gym + gtest
|
|
|
|
else
|
|
|
|
c Call dcstep to update stx, sty, and to compute the new step.
|
|
|
|
call dcstep(stx,fx,gx,sty,fy,gy,stp,f,g,
|
|
+ brackt,stmin,stmax)
|
|
|
|
endif
|
|
|
|
c Decide if a bisection step is needed.
|
|
|
|
if (brackt) then
|
|
if (abs(sty-stx) .ge. p66*width1) stp = stx + p5*(sty - stx)
|
|
width1 = width
|
|
width = abs(sty-stx)
|
|
endif
|
|
|
|
c Set the minimum and maximum steps allowed for stp.
|
|
|
|
if (brackt) then
|
|
stmin = min(stx,sty)
|
|
stmax = max(stx,sty)
|
|
else
|
|
stmin = stp + xtrapl*(stp - stx)
|
|
stmax = stp + xtrapu*(stp - stx)
|
|
endif
|
|
|
|
c Force the step to be within the bounds stpmax and stpmin.
|
|
|
|
stp = max(stp,stpmin)
|
|
stp = min(stp,stpmax)
|
|
|
|
c If further progress is not possible, let stp be the best
|
|
c point obtained during the search.
|
|
|
|
if (brackt .and. (stp .le. stmin .or. stp .ge. stmax)
|
|
+ .or. (brackt .and. stmax-stmin .le. xtol*stmax)) stp = stx
|
|
|
|
c Obtain another function and derivative.
|
|
|
|
task = 'FG'
|
|
|
|
1000 continue
|
|
|
|
c Save local variables.
|
|
|
|
if (brackt) then
|
|
isave(1) = 1
|
|
else
|
|
isave(1) = 0
|
|
endif
|
|
isave(2) = stage
|
|
dsave(1) = ginit
|
|
dsave(2) = gtest
|
|
dsave(3) = gx
|
|
dsave(4) = gy
|
|
dsave(5) = finit
|
|
dsave(6) = fx
|
|
dsave(7) = fy
|
|
dsave(8) = stx
|
|
dsave(9) = sty
|
|
dsave(10) = stmin
|
|
dsave(11) = stmax
|
|
dsave(12) = width
|
|
dsave(13) = width1
|
|
|
|
return
|
|
end
|
|
|
|
c====================== The end of dcsrch ==============================
|
|
|
|
subroutine dcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,brackt,
|
|
+ stpmin,stpmax)
|
|
logical brackt
|
|
double precision stx,fx,dx,sty,fy,dy,stp,fp,dp,stpmin,stpmax
|
|
c **********
|
|
c
|
|
c Subroutine dcstep
|
|
c
|
|
c This subroutine computes a safeguarded step for a search
|
|
c procedure and updates an interval that contains a step that
|
|
c satisfies a sufficient decrease and a curvature condition.
|
|
c
|
|
c The parameter stx contains the step with the least function
|
|
c value. If brackt is set to .true. then a minimizer has
|
|
c been bracketed in an interval with endpoints stx and sty.
|
|
c The parameter stp contains the current step.
|
|
c The subroutine assumes that if brackt is set to .true. then
|
|
c
|
|
c min(stx,sty) < stp < max(stx,sty),
|
|
c
|
|
c and that the derivative at stx is negative in the direction
|
|
c of the step.
|
|
c
|
|
c The subroutine statement is
|
|
c
|
|
c subroutine dcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,brackt,
|
|
c stpmin,stpmax)
|
|
c
|
|
c where
|
|
c
|
|
c stx is a double precision variable.
|
|
c On entry stx is the best step obtained so far and is an
|
|
c endpoint of the interval that contains the minimizer.
|
|
c On exit stx is the updated best step.
|
|
c
|
|
c fx is a double precision variable.
|
|
c On entry fx is the function at stx.
|
|
c On exit fx is the function at stx.
|
|
c
|
|
c dx is a double precision variable.
|
|
c On entry dx is the derivative of the function at
|
|
c stx. The derivative must be negative in the direction of
|
|
c the step, that is, dx and stp - stx must have opposite
|
|
c signs.
|
|
c On exit dx is the derivative of the function at stx.
|
|
c
|
|
c sty is a double precision variable.
|
|
c On entry sty is the second endpoint of the interval that
|
|
c contains the minimizer.
|
|
c On exit sty is the updated endpoint of the interval that
|
|
c contains the minimizer.
|
|
c
|
|
c fy is a double precision variable.
|
|
c On entry fy is the function at sty.
|
|
c On exit fy is the function at sty.
|
|
c
|
|
c dy is a double precision variable.
|
|
c On entry dy is the derivative of the function at sty.
|
|
c On exit dy is the derivative of the function at the exit sty.
|
|
c
|
|
c stp is a double precision variable.
|
|
c On entry stp is the current step. If brackt is set to .true.
|
|
c then on input stp must be between stx and sty.
|
|
c On exit stp is a new trial step.
|
|
c
|
|
c fp is a double precision variable.
|
|
c On entry fp is the function at stp
|
|
c On exit fp is unchanged.
|
|
c
|
|
c dp is a double precision variable.
|
|
c On entry dp is the the derivative of the function at stp.
|
|
c On exit dp is unchanged.
|
|
c
|
|
c brackt is an logical variable.
|
|
c On entry brackt specifies if a minimizer has been bracketed.
|
|
c Initially brackt must be set to .false.
|
|
c On exit brackt specifies if a minimizer has been bracketed.
|
|
c When a minimizer is bracketed brackt is set to .true.
|
|
c
|
|
c stpmin is a double precision variable.
|
|
c On entry stpmin is a lower bound for the step.
|
|
c On exit stpmin is unchanged.
|
|
c
|
|
c stpmax is a double precision variable.
|
|
c On entry stpmax is an upper bound for the step.
|
|
c On exit stpmax is unchanged.
|
|
c
|
|
c MINPACK-1 Project. June 1983
|
|
c Argonne National Laboratory.
|
|
c Jorge J. More' and David J. Thuente.
|
|
c
|
|
c MINPACK-2 Project. October 1993.
|
|
c Argonne National Laboratory and University of Minnesota.
|
|
c Brett M. Averick and Jorge J. More'.
|
|
c
|
|
c **********
|
|
double precision zero,p66,two,three
|
|
parameter(zero=0.0d0,p66=0.66d0,two=2.0d0,three=3.0d0)
|
|
|
|
double precision gamma,p,q,r,s,sgnd,stpc,stpf,stpq,theta
|
|
|
|
sgnd = dp*(dx/abs(dx))
|
|
|
|
c First case: A higher function value. The minimum is bracketed.
|
|
c If the cubic step is closer to stx than the quadratic step, the
|
|
c cubic step is taken, otherwise the average of the cubic and
|
|
c quadratic steps is taken.
|
|
|
|
if (fp .gt. fx) then
|
|
theta = three*(fx - fp)/(stp - stx) + dx + dp
|
|
s = max(abs(theta),abs(dx),abs(dp))
|
|
gamma = s*sqrt((theta/s)**2 - (dx/s)*(dp/s))
|
|
if (stp .lt. stx) gamma = -gamma
|
|
p = (gamma - dx) + theta
|
|
q = ((gamma - dx) + gamma) + dp
|
|
r = p/q
|
|
stpc = stx + r*(stp - stx)
|
|
stpq = stx + ((dx/((fx - fp)/(stp - stx) + dx))/two)*
|
|
+ (stp - stx)
|
|
if (abs(stpc-stx) .lt. abs(stpq-stx)) then
|
|
stpf = stpc
|
|
else
|
|
stpf = stpc + (stpq - stpc)/two
|
|
endif
|
|
brackt = .true.
|
|
|
|
c Second case: A lower function value and derivatives of opposite
|
|
c sign. The minimum is bracketed. If the cubic step is farther from
|
|
c stp than the secant step, the cubic step is taken, otherwise the
|
|
c secant step is taken.
|
|
|
|
else if (sgnd .lt. zero) then
|
|
theta = three*(fx - fp)/(stp - stx) + dx + dp
|
|
s = max(abs(theta),abs(dx),abs(dp))
|
|
gamma = s*sqrt((theta/s)**2 - (dx/s)*(dp/s))
|
|
if (stp .gt. stx) gamma = -gamma
|
|
p = (gamma - dp) + theta
|
|
q = ((gamma - dp) + gamma) + dx
|
|
r = p/q
|
|
stpc = stp + r*(stx - stp)
|
|
stpq = stp + (dp/(dp - dx))*(stx - stp)
|
|
if (abs(stpc-stp) .gt. abs(stpq-stp)) then
|
|
stpf = stpc
|
|
else
|
|
stpf = stpq
|
|
endif
|
|
brackt = .true.
|
|
|
|
c Third case: A lower function value, derivatives of the same sign,
|
|
c and the magnitude of the derivative decreases.
|
|
|
|
else if (abs(dp) .lt. abs(dx)) then
|
|
|
|
c The cubic step is computed only if the cubic tends to infinity
|
|
c in the direction of the step or if the minimum of the cubic
|
|
c is beyond stp. Otherwise the cubic step is defined to be the
|
|
c secant step.
|
|
|
|
theta = three*(fx - fp)/(stp - stx) + dx + dp
|
|
s = max(abs(theta),abs(dx),abs(dp))
|
|
|
|
c The case gamma = 0 only arises if the cubic does not tend
|
|
c to infinity in the direction of the step.
|
|
|
|
gamma = s*sqrt(max(zero,(theta/s)**2-(dx/s)*(dp/s)))
|
|
if (stp .gt. stx) gamma = -gamma
|
|
p = (gamma - dp) + theta
|
|
q = (gamma + (dx - dp)) + gamma
|
|
r = p/q
|
|
if (r .lt. zero .and. gamma .ne. zero) then
|
|
stpc = stp + r*(stx - stp)
|
|
else if (stp .gt. stx) then
|
|
stpc = stpmax
|
|
else
|
|
stpc = stpmin
|
|
endif
|
|
stpq = stp + (dp/(dp - dx))*(stx - stp)
|
|
|
|
if (brackt) then
|
|
|
|
c A minimizer has been bracketed. If the cubic step is
|
|
c closer to stp than the secant step, the cubic step is
|
|
c taken, otherwise the secant step is taken.
|
|
|
|
if (abs(stpc-stp) .lt. abs(stpq-stp)) then
|
|
stpf = stpc
|
|
else
|
|
stpf = stpq
|
|
endif
|
|
if (stp .gt. stx) then
|
|
stpf = min(stp+p66*(sty-stp),stpf)
|
|
else
|
|
stpf = max(stp+p66*(sty-stp),stpf)
|
|
endif
|
|
else
|
|
|
|
c A minimizer has not been bracketed. If the cubic step is
|
|
c farther from stp than the secant step, the cubic step is
|
|
c taken, otherwise the secant step is taken.
|
|
|
|
if (abs(stpc-stp) .gt. abs(stpq-stp)) then
|
|
stpf = stpc
|
|
else
|
|
stpf = stpq
|
|
endif
|
|
stpf = min(stpmax,stpf)
|
|
stpf = max(stpmin,stpf)
|
|
endif
|
|
|
|
c Fourth case: A lower function value, derivatives of the same sign,
|
|
c and the magnitude of the derivative does not decrease. If the
|
|
c minimum is not bracketed, the step is either stpmin or stpmax,
|
|
c otherwise the cubic step is taken.
|
|
|
|
else
|
|
if (brackt) then
|
|
theta = three*(fp - fy)/(sty - stp) + dy + dp
|
|
s = max(abs(theta),abs(dy),abs(dp))
|
|
gamma = s*sqrt((theta/s)**2 - (dy/s)*(dp/s))
|
|
if (stp .gt. sty) gamma = -gamma
|
|
p = (gamma - dp) + theta
|
|
q = ((gamma - dp) + gamma) + dy
|
|
r = p/q
|
|
stpc = stp + r*(sty - stp)
|
|
stpf = stpc
|
|
else if (stp .gt. stx) then
|
|
stpf = stpmax
|
|
else
|
|
stpf = stpmin
|
|
endif
|
|
endif
|
|
|
|
c Update the interval which contains a minimizer.
|
|
|
|
if (fp .gt. fx) then
|
|
sty = stp
|
|
fy = fp
|
|
dy = dp
|
|
else
|
|
if (sgnd .lt. zero) then
|
|
sty = stx
|
|
fy = fx
|
|
dy = dx
|
|
endif
|
|
stx = stp
|
|
fx = fp
|
|
dx = dp
|
|
endif
|
|
|
|
c Compute the new step.
|
|
|
|
stp = stpf
|
|
|
|
return
|
|
end
|
|
|