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!
! 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
!
! 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
! Allocate dynamic arrays
allocate ( nbd(n), x(n), l(n), u(n), g(n) )
allocate ( iwa(3*n) )
allocate ( wa(2*m*n + 5*n + 11*m*m + 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'
! 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
end program driver
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