c c L-BFGS-B is released under the “New BSD License” (aka “Modified BSD License” c or “3-clause license”) c Please read attached file License.txt c c DRIVER 1 in Fortran 77 c -------------------------------------------------------------- c SIMPLE DRIVER FOR L-BFGS-B (version 3.0) c -------------------------------------------------------------- c c L-BFGS-B is a code for solving large nonlinear optimization c problems with simple bounds on the variables. c c The code can also be used for unconstrained problems and is c as efficient for these problems as the earlier limited memory c code L-BFGS. c c This is the simplest driver in the package. It uses all the c default settings of the code. 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 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 March 2011 (latest revision) c Optimization Center at Northwestern University c Instituto Tecnologico Autonomo de Mexico c c Jorge Nocedal and Jose Luis Morales, Remark on "Algorithm 778: c L-BFGS-B: Fortran Subroutines for Large-Scale Bound Constrained c Optimization" (2011). To appear in ACM Transactions on c Mathematical Software, c -------------------------------------------------------------- c DESCRIPTION OF THE VARIABLES IN L-BFGS-B c -------------------------------------------------------------- c c n is an INTEGER variable that must be set by the user to the c number of variables. It is not altered by the routine. c c m is an INTEGER variable that must be set by the user to the c number of corrections used in the limited memory matrix. c It is not altered by the routine. Values of m < 3 are c not recommended, and large values of m can result in excessive c computing time. The range 3 <= m <= 20 is recommended. c c x is a DOUBLE PRECISION array of length n. On initial entry c it must be set by the user to the values of the initial c estimate of the solution vector. Upon successful exit, it c contains the values of the variables at the best point c found (usually an approximate solution). c c l is a DOUBLE PRECISION array of length n that must be set by c the user to the values of the lower bounds on the variables. If c the i-th variable has no lower bound, l(i) need not be defined. c c u is a DOUBLE PRECISION array of length n that must be set by c the user to the values of the upper bounds on the variables. If c the i-th variable has no upper bound, u(i) need not be defined. c c nbd is an INTEGER array of dimension n that must be set by the c user to the type of bounds imposed on the variables: 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, c 3 if x(i) has only an upper bound. c c f is a DOUBLE PRECISION variable. If the routine setulb returns c with task(1:2)= 'FG', then f must be set by the user to c contain the value of the function at the point x. c c g is a DOUBLE PRECISION array of length n. If the routine setulb c returns with taskb(1:2)= 'FG', then g must be set by the user to c contain the components of the gradient at the point x. c c factr is a DOUBLE PRECISION variable that must be set by the user. c It is a tolerance in the termination test for the algorithm. c The iteration will stop when c c (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch c c where epsmch is the machine precision which is automatically c generated by the code. Typical values for factr on a computer c with 15 digits of accuracy in double precision are: c factr=1.d+12 for low accuracy; c 1.d+7 for moderate accuracy; c 1.d+1 for extremely high accuracy. c The user can suppress this termination test by setting factr=0. c c pgtol is a double precision variable. c On entry pgtol >= 0 is specified by the user. The iteration c will stop when c c max{|proj g_i | i = 1, ..., n} <= pgtol c c where pg_i is the ith component of the projected gradient. c The user can suppress this termination test by setting pgtol=0. c c wa is a DOUBLE PRECISION array of length c (2mmax + 5)nmax + 11mmax^2 + 8mmax used as workspace. c This array must not be altered by the user. c c iwa is an INTEGER array of length 3nmax used as c workspace. This array must not be altered by the user. c c task is a CHARACTER string of length 60. c On first entry, it must be set to 'START'. c On a return with task(1:2)='FG', the user must evaluate the c function f and gradient g at the returned value of x. c On a return with task(1:5)='NEW_X', an iteration of the c algorithm has concluded, and f and g contain f(x) and g(x) c respectively. The user can decide whether to continue or stop c the iteration. c When c task(1:4)='CONV', the termination test in L-BFGS-B has been c satisfied; c task(1:4)='ABNO', the routine has terminated abnormally c without being able to satisfy the termination conditions, c x contains the best approximation found, c f and g contain f(x) and g(x) respectively; c task(1:5)='ERROR', the routine has detected an error in the c input parameters; c On exit with task = 'CONV', 'ABNO' or 'ERROR', the variable task c contains additional information that the user can print. c This array should not be altered unless the user wants to c stop the run for some reason. See driver2 or driver3 c for a detailed explanation on how to stop the run c by assigning task(1:4)='STOP' in the driver. 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 0100 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 csave is a CHARACTER working array of length 60. c c lsave is a LOGICAL working array of dimension 4. c On exit with task = 'NEW_X', the following information is c available: c lsave(1) = .true. the initial x did not satisfy the bounds; c lsave(2) = .true. the problem contains bounds; c lsave(3) = .true. each variable has upper and lower bounds. c c isave is an INTEGER working array of dimension 44. c On exit with task = 'NEW_X', it contains information that c the user may want to access: c isave(30) = the current iteration number; c isave(34) = the total number of function and gradient c evaluations; c isave(36) = the number of function value or gradient c evaluations in the current iteration; c isave(38) = the number of free variables in the current c iteration; c isave(39) = the number of active constraints at the current c iteration; c c see the subroutine setulb.f for a description of other c information contained in isave c c dsave is a DOUBLE PRECISION working array of dimension 29. c On exit with task = 'NEW_X', it contains information that c the user may want to access: c dsave(2) = the value of f at the previous iteration; c dsave(5) = the machine precision epsmch generated by the code; c dsave(13) = the infinity norm of the projected gradient; c c see the subroutine setulb.f for a description of other c information contained in dsave c c -------------------------------------------------------------- c END OF THE DESCRIPTION OF THE VARIABLES IN L-BFGS-B c -------------------------------------------------------------- program driver c This simple driver demonstrates how to call the L-BFGS-B code to c solve a sample problem (the extended Rosenbrock function c subject to bounds on the variables). The dimension n of this c problem is variable. integer nmax, mmax parameter (nmax=1024, mmax=17) c nmax is the dimension of the largest problem to be solved. c mmax is the maximum number of limited memory corrections. c Declare the variables needed by the code. c A description of all these variables is given at the end of c the driver. character*60 task, csave logical lsave(4) integer n, m, iprint, + nbd(nmax), iwa(3*nmax), isave(44) double precision f, factr, pgtol, + x(nmax), l(nmax), u(nmax), g(nmax), dsave(29), + wa(2*mmax*nmax + 5*nmax + 11*mmax*mmax + 8*mmax) c Declare a few additional variables for this sample problem. double precision t1, t2 integer i c We wish to have output at every iteration. iprint = 1 c We specify the tolerances in the stopping criteria. factr=1.0d+7 pgtol=1.0d-5 c We specify the dimension n of the sample problem and the number c m of limited memory corrections stored. (n and m should not c exceed the limits nmax and mmax respectively.) n=25 m=5 c We now provide nbd which defines the bounds on the variables: c l specifies the lower bounds, c u specifies the upper bounds. c First set bounds on the odd-numbered variables. do 10 i=1,n,2 nbd(i)=2 l(i)=1.0d0 u(i)=1.0d2 10 continue c 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 c 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.)',/) c We start the iteration by initializing task. c task = 'START' c ------- the beginning of the loop ---------- 111 continue c 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 c the minimization routine has returned to request the c function f and gradient g values at the current x. c Compute function value f for the sample problem. 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 c 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 c go back to the minimization routine. goto 111 endif c if (task(1:5) .eq. 'NEW_X') goto 111 c the minimization routine has returned with a new iterate, c and we have opted to continue the iteration. c ---------- the end of the loop ------------- c If task is neither FG nor NEW_X we terminate execution. stop end c======================= The end of driver1 ============================