L-BFGS-B/driver1_8f90-example.html

 !> \file driver1.f90


 !

 !  L-BFGS-B is released under the "New BSD License" (aka "Modified BSD License"

 !  or "3-clause license")

 !  Please read attached file License.txt

 !

 !

 !                      DRIVER1 in Fortran 90

 !     --------------------------------------------------------------

 !

 !        L-BFGS-B is a code for solving large nonlinear optimization

 !             problems with simple bounds on the variables.

 !

 !        The code can also be used for unconstrained problems and is

 !        as efficient for these problems as the earlier limited memory

 !                          code L-BFGS.

 !

 !        This is the simplest driver in the package. It uses all the

 !                    default settings of the code.

 !

 !

 !     References:

 !

 !        [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited

 !        memory algorithm for bound constrained optimization'',

 !        SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.

 !

 !        [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN

 !        Subroutines for Large Scale Bound Constrained Optimization''

 !        Tech. Report, NAM-11, EECS Department, Northwestern University,

 !        1994.

 !

 !

 !          (Postscript files of these papers are available via anonymous

 !           ftp to eecs.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)

 !

 !                              *  *  *

 !

 !         March 2011   (latest revision)

 !         Optimization Center at Northwestern University

 !         Instituto Tecnologico Autonomo de Mexico

 !

 !         Jorge Nocedal and Jose Luis Morales

 !

 !     --------------------------------------------------------------

 !             DESCRIPTION OF THE VARIABLES IN L-BFGS-B

 !     --------------------------------------------------------------

 !

 !     n is an INTEGER variable that must be set by the user to the

 !       number of variables.  It is not altered by the routine.

 !

 !     m is an INTEGER variable that must be set by the user to the

 !       number of corrections used in the limited memory matrix.

 !       It is not altered by the routine.  Values of m < 3  are

 !       not recommended, and large values of m can result in excessive

 !       computing time. The range  3 <= m <= 20 is recommended.

 !

 !     x is a DOUBLE PRECISION array of length n.  On initial entry

 !       it must be set by the user to the values of the initial

 !       estimate of the solution vector.  Upon successful exit, it

 !       contains the values of the variables at the best point

 !       found (usually an approximate solution).

 !

 !     l is a DOUBLE PRECISION array of length n that must be set by

 !       the user to the values of the lower bounds on the variables. If

 !       the i-th variable has no lower bound, l(i) need not be defined.

 !

 !     u is a DOUBLE PRECISION array of length n that must be set by

 !       the user to the values of the upper bounds on the variables. If

 !       the i-th variable has no upper bound, u(i) need not be defined.

 !

 !     nbd is an INTEGER array of dimension n that must be set by the

 !       user to the type of bounds imposed on the variables:

 !       nbd(i)=0 if x(i) is unbounded,

 !              1 if x(i) has only a lower bound,

 !              2 if x(i) has both lower and upper bounds,

 !              3 if x(i) has only an upper bound.

 !

 !     f is a DOUBLE PRECISION variable.  If the routine setulb returns

 !       with task(1:2)= 'FG', then f must be set by the user to

 !       contain the value of the function at the point x.

 !

 !     g is a DOUBLE PRECISION array of length n.  If the routine setulb

 !       returns with taskb(1:2)= 'FG', then g must be set by the user to

 !       contain the components of the gradient at the point x.

 !

 !     factr is a DOUBLE PRECISION variable that must be set by the user.

 !       It is a tolerance in the termination test for the algorithm.

 !       The iteration will stop when

 !

 !        (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch

 !

 !       where epsmch is the machine precision which is automatically

 !       generated by the code. Typical values for factr on a computer

 !       with 15 digits of accuracy in double precision are:

 !       factr=1.d+12 for low accuracy;

 !             1.d+7  for moderate accuracy;

 !             1.d+1  for extremely high accuracy.

 !       The user can suppress this termination test by setting factr=0.

 !

 !     pgtol is a double precision variable.

 !       On entry pgtol >= 0 is specified by the user.  The iteration

 !         will stop when

 !

 !                 max{|proj g_i | i = 1, ..., n} <= pgtol

 !

 !         where pg_i is the ith component of the projected gradient.

 !       The user can suppress this termination test by setting pgtol=0.

 !

 !     wa is a DOUBLE PRECISION  array of length

 !       (2mmax + 5)nmax + 11mmax^2 + 8mmax used as workspace.

 !       This array must not be altered by the user.

 !

 !     iwa is an INTEGER  array of length 3nmax used as

 !       workspace. This array must not be altered by the user.

 !

 !     task is a CHARACTER string of length 60.

 !       On first entry, it must be set to 'START'.

 !       On a return with task(1:2)='FG', the user must evaluate the

 !         function f and gradient g at the returned value of x.

 !       On a return with task(1:5)='NEW_X', an iteration of the

 !         algorithm has concluded, and f and g contain f(x) and g(x)

 !         respectively.  The user can decide whether to continue or stop

 !         the iteration.

 !       When

 !         task(1:4)='CONV', the termination test in L-BFGS-B has been

 !           satisfied;

 !         task(1:4)='ABNO', the routine has terminated abnormally

 !           without being able to satisfy the termination conditions,

 !           x contains the best approximation found,

 !           f and g contain f(x) and g(x) respectively;

 !         task(1:5)='ERROR', the routine has detected an error in the

 !           input parameters;

 !       On exit with task = 'CONV', 'ABNO' or 'ERROR', the variable task

 !         contains additional information that the user can print.

 !       This array should not be altered unless the user wants to

 !          stop the run for some reason.  See driver2 or driver3

 !          for a detailed explanation on how to stop the run

 !          by assigning task(1:4)='STOP' in the driver.

 !

 !     iprint is an INTEGER variable that must be set by the user.

 !       It controls the frequency and type of output generated:

 !        iprint<0    no output is generated;

 !        iprint=0    print only one line at the last iteration;

 !        0<iprint<99 print also f and |proj g| every iprint iterations;

 !        iprint=99   print details of every iteration except n-vectors;

 !        iprint=100  print also the changes of active set and final x;

 !        iprint>100  print details of every iteration including x and g;

 !       When iprint > 0, the file iterate.dat will be created to

 !                        summarize the iteration.

 !

 !     csave  is a CHARACTER working array of length 60.

 !

 !     lsave is a LOGICAL working array of dimension 4.

 !       On exit with task = 'NEW_X', the following information is

 !         available:

 !       lsave(1) = .true.  the initial x did not satisfy the bounds;

 !       lsave(2) = .true.  the problem contains bounds;

 !       lsave(3) = .true.  each variable has upper and lower bounds.

 !

 !     isave is an INTEGER working array of dimension 44.

 !       On exit with task = 'NEW_X', it contains information that

 !       the user may want to access:

 !         isave(30) = the current iteration number;

 !         isave(34) = the total number of function and gradient

 !                         evaluations;

 !         isave(36) = the number of function value or gradient

 !                                  evaluations in the current iteration;

 !         isave(38) = the number of free variables in the current

 !                         iteration;

 !         isave(39) = the number of active constraints at the current

 !                         iteration;

 !

 !         see the subroutine setulb.f for a description of other

 !         information contained in isave

 !

 !     dsave is a DOUBLE PRECISION working array of dimension 29.

 !       On exit with task = 'NEW_X', it contains information that

 !         the user may want to access:

 !         dsave(2)  = the value of f at the previous iteration;

 !         dsave(5)  = the machine precision epsmch generated by the code;

 !         dsave(13) = the infinity norm of the projected gradient;

 !

 !         see the subroutine setulb.f for a description of other

 !         information contained in dsave

 !

 !     --------------------------------------------------------------

 !           END OF THE DESCRIPTION OF THE VARIABLES IN L-BFGS-B

 !     --------------------------------------------------------------

 !

       program driver

       use json_writer, only: json_write_aggregate

 !

 !     This simple driver demonstrates how to call the L-BFGS-B code to

 !       solve a sample problem (the extended Rosenbrock function

 !       subject to bounds on the variables). The dimension n of this

 !       problem is variable.


       implicit none

 !

 !     Declare variables and parameters needed by the code.

 !       Note thar we wish to have output at every iteration.

 !          iprint=1

 !

 !       We also specify the tolerances in the stopping criteria.

 !          factr  = 1.0d+7, pgtol  = 1.0d-5

 !

 !       A description of all these variables is given at the beginning

 !       of  the driver

 !

       integer,  parameter    :: n = 25, m = 5, iprint = 1

       integer,  parameter    :: dp = kind(1.0d0)

       real(dp), parameter    :: factr  = 1.0d+7, pgtol  = 1.0d-5

 !

       character(len=60)      :: task, csave

       logical                :: lsave(4)

       integer                :: isave(44)

       real(dp)               :: f

       real(dp)               :: dsave(29)

       integer,  allocatable  :: nbd(:), iwa(:)

       real(dp), allocatable  :: x(:), l(:), u(:), g(:), wa(:)


 !     Declare a few additional variables for this sample problem


       real(dp)               :: t1, t2

       integer                :: i


 !     Test instrumentation: dump per-iteration state to JSON when the

 !     environment variable LBFGSB_JSON_OUTPUT is set. No-op otherwise.

       character(len=512)     :: lbfgsb_json

       logical                :: json_active


 !     Allocate dynamic arrays


       allocate ( nbd(n), x(n), l(n), u(n), g(n) )

       allocate ( iwa(3*n) )

       allocate ( wa(2*m*n + 11*m*m + 5*n + 8*m) )

 !

       do 10 i=1, n, 2

          nbd(i) = 2

          l(i)   = 1.0d0

          u(i)   = 1.0d2

   10  continue


 !     Next set bounds on the even-numbered variables.


       do 12 i=2, n, 2

          nbd(i) =  2

          l(i)   = -1.0d2

          u(i)   =  1.0d2

   12   continue


 !     We now define the starting point.


       do 14 i=1, n

          x(i) = 3.0d0

   14  continue


       write (6,16)

   16  format(/,5x, 'Solving sample problem.', &

              /,5x, ' (f = 0.0 at the optimal solution.)',/)


 !     We start the iteration by initializing task.


       task = 'START'


 !     Test instrumentation: enabled when LBFGSB_JSON_OUTPUT is set.

       call get_environment_variable('LBFGSB_JSON_OUTPUT', lbfgsb_json)

       json_active = (len_trim(lbfgsb_json) > 0)


 !     The beginning of the loop


       do while(task(1:2).eq.'FG'.or.task.eq.'NEW_X'.or. &

                task.eq.'START')


 !     This is the call to the L-BFGS-B code.


          call setulb ( n, m, x, l, u, nbd, f, g, factr, pgtol, &

                        wa, iwa, task, iprint,&

                        csave, lsave, isave, dsave )


          if (task(1:2) .eq. 'FG') then


             f=.25d0*( x(1)-1.d0 )**2

             do 20 i=2, n

                f = f + ( x(i)-x(i-1 )**2 )**2

  20         continue

             f = 4.d0*f


 !     Compute gradient g for the sample problem.


             t1   = x(2) - x(1)**2

             g(1) = 2.d0*(x(1) - 1.d0) - 1.6d1*x(1)*t1

             do 22 i=2, n-1

                t2   = t1

                t1   = x(i+1) - x(i)**2

                g(i) = 8.d0*t2 - 1.6d1*x(i)*t1

  22         continue

             g(n) = 8.d0*t1


          end if

       end do


 !     end of loop do while


       if (json_active) &

          call json_write_aggregate(trim(lbfgsb_json), task, f, dsave(13), n, x)


       end program driver


setulb
subroutine setulb(n, m, x, l, u, nbd, f, g, factr, pgtol, wa, iwa, task, iprint, csave, lsave, isave, dsave)
This subroutine partitions the working arrays wa and iwa, and then uses the limited memory BFGS metho...
Definition: setulb.f:190