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lbfgsb.f
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lbfgsb.f
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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=========== L-BFGS-B (version 3.0. April 25, 2011 ===================
c
c This is a modified version of L-BFGS-B. Minor changes in the updated
c code appear preceded by a line comment as follows
c
c c-jlm-jn
c
c Major changes are described in the accompanying paper:
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 The paper describes an improvement and a correction to Algorithm 778.
c It is shown that the performance of the algorithm can be improved
c significantly by making a relatively simple modication to the subspace
c minimization phase. The correction concerns an error caused by the use
c of routine dpmeps to estimate machine precision.
c
c The total work space **wa** required by the new version is
c
c 2*m*n + 11m*m + 5*n + 8*m
c
c the old version required
c
c 2*m*n + 12m*m + 4*n + 12*m
c
c
c J. Nocedal Department of Electrical Engineering and
c Computer Science.
c Northwestern University. Evanston, IL. USA
c
c
c J.L Morales Departamento de Matematicas,
c Instituto Tecnologico Autonomo de Mexico
c Mexico D.F. Mexico.
c
c March 2011
c
c=============================================================================
subroutine setulb(n, m, x, l, u, nbd, f, g, factr, pgtol, wa, iwa,
+ task, iprint, csave, lsave, isave, dsave)
character*60 task, csave
logical lsave(4)
integer n, m, iprint,
+ nbd(n), iwa(3*n), isave(44)
double precision f, factr, pgtol, x(n), l(n), u(n), g(n),
c
c-jlm-jn
+ wa(2*m*n + 5*n + 11*m*m + 8*m), dsave(29)
c ************
c
c Subroutine setulb
c
c This subroutine partitions the working arrays wa and iwa, and
c then uses the limited memory BFGS method to solve the bound
c constrained optimization problem by calling mainlb.
c (The direct method will be used in the subspace minimization.)
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 x is a double precision array of dimension n.
c On entry x is an approximation to the solution.
c On exit x is the current approximation.
c
c l is a double precision array of dimension n.
c On entry l is the lower bound on 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 on 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 f is a double precision variable.
c On first entry f is unspecified.
c On final exit f is the value of the function at x.
c
c g is a double precision array of dimension n.
c On first entry g is unspecified.
c On final exit g is the value of the gradient at x.
c
c factr is a double precision variable.
c On entry factr >= 0 is specified by the user. The iteration
c 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: 1.d+12 for
c low accuracy; 1.d+7 for moderate accuracy; 1.d+1 for extremely
c high accuracy.
c On exit factr is unchanged.
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 On exit pgtol is unchanged.
c
c wa is a double precision working array of length
c (2mmax + 5)nmax + 12mmax^2 + 12mmax.
c
c iwa is an integer working array of length 3nmax.
c
c task is a working string of characters of length 60 indicating
c the current job when entering and quitting this subroutine.
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 csave is a working string of characters 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 If lsave(1) = .true. then the initial X has been replaced by
c its projection in the feasible set;
c If lsave(2) = .true. then the problem is constrained;
c If lsave(3) = .true. then each variable has upper and lower
c bounds;
c
c isave is an integer working array of dimension 44.
c On exit with 'task' = NEW_X, the following information is
c available:
c isave(22) = the total number of intervals explored in the
c search of Cauchy points;
c isave(26) = the total number of skipped BFGS updates before
c the current iteration;
c isave(30) = the number of current iteration;
c isave(31) = the total number of BFGS updates prior the current
c iteration;
c isave(33) = the number of intervals explored in the search of
c Cauchy point in the current iteration;
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 if isave(37) = 0 then the subspace argmin is within the box;
c if isave(37) = 1 then the subspace argmin is beyond the box;
c isave(38) = the number of free variables in the current
c iteration;
c isave(39) = the number of active constraints in the current
c iteration;
c n + 1 - isave(40) = the number of variables leaving the set of
c active constraints in the current iteration;
c isave(41) = the number of variables entering the set of active
c constraints in the current iteration.
c
c dsave is a double precision working array of dimension 29.
c On exit with 'task' = NEW_X, the following information is
c available:
c dsave(1) = current 'theta' in the BFGS matrix;
c dsave(2) = f(x) in the previous iteration;
c dsave(3) = factr*epsmch;
c dsave(4) = 2-norm of the line search direction vector;
c dsave(5) = the machine precision epsmch generated by the code;
c dsave(7) = the accumulated time spent on searching for
c Cauchy points;
c dsave(8) = the accumulated time spent on
c subspace minimization;
c dsave(9) = the accumulated time spent on line search;
c dsave(11) = the slope of the line search function at
c the current point of line search;
c dsave(12) = the maximum relative step length imposed in
c line search;
c dsave(13) = the infinity norm of the projected gradient;
c dsave(14) = the relative step length in the line search;
c dsave(15) = the slope of the line search function at
c the starting point of the line search;
c dsave(16) = the square of the 2-norm of the line search
c direction vector.
c
c Subprograms called:
c
c L-BFGS-B Library ... mainlb.
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: 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 ************
c-jlm-jn
integer lws,lr,lz,lt,ld,lxp,lwa,
+ lwy,lsy,lss,lwt,lwn,lsnd
if (task .eq. 'START') then
isave(1) = m*n
isave(2) = m**2
isave(3) = 4*m**2
isave(4) = 1 ! ws m*n
isave(5) = isave(4) + isave(1) ! wy m*n
isave(6) = isave(5) + isave(1) ! wsy m**2
isave(7) = isave(6) + isave(2) ! wss m**2
isave(8) = isave(7) + isave(2) ! wt m**2
isave(9) = isave(8) + isave(2) ! wn 4*m**2
isave(10) = isave(9) + isave(3) ! wsnd 4*m**2
isave(11) = isave(10) + isave(3) ! wz n
isave(12) = isave(11) + n ! wr n
isave(13) = isave(12) + n ! wd n
isave(14) = isave(13) + n ! wt n
isave(15) = isave(14) + n ! wxp n
isave(16) = isave(15) + n ! wa 8*m
endif
lws = isave(4)
lwy = isave(5)
lsy = isave(6)
lss = isave(7)
lwt = isave(8)
lwn = isave(9)
lsnd = isave(10)
lz = isave(11)
lr = isave(12)
ld = isave(13)
lt = isave(14)
lxp = isave(15)
lwa = isave(16)
call mainlb(n,m,x,l,u,nbd,f,g,factr,pgtol,
+ wa(lws),wa(lwy),wa(lsy),wa(lss), wa(lwt),
+ wa(lwn),wa(lsnd),wa(lz),wa(lr),wa(ld),wa(lt),wa(lxp),
+ wa(lwa),
+ iwa(1),iwa(n+1),iwa(2*n+1),task,iprint,
+ csave,lsave,isave(22),dsave)
return
end
c======================= The end of setulb =============================
subroutine mainlb(n, m, x, l, u, nbd, f, g, factr, pgtol, ws, wy,
+ sy, ss, wt, wn, snd, z, r, d, t, xp, wa,
+ index, iwhere, indx2, task,
+ iprint, csave, lsave, isave, dsave)
implicit none
character*60 task, csave
logical lsave(4)
integer n, m, iprint, nbd(n), index(n),
+ iwhere(n), indx2(n), isave(23)
double precision f, factr, pgtol,
+ x(n), l(n), u(n), g(n), z(n), r(n), d(n), t(n),
c-jlm-jn
+ xp(n),
+ wa(8*m),
+ ws(n, m), wy(n, m), sy(m, m), ss(m, m),
+ wt(m, m), wn(2*m, 2*m), snd(2*m, 2*m), dsave(29)
c ************
c
c Subroutine mainlb
c
c This subroutine solves bound constrained optimization problems by
c using the compact formula of the limited memory BFGS updates.
c
c n is an integer variable.
c On entry n is the number of variables.
c On exit n is unchanged.
c
c m is an integer variable.
c On entry m is the maximum number of variable metric
c corrections allowed in the limited memory matrix.
c On exit m is unchanged.
c
c x is a double precision array of dimension n.
c On entry x is an approximation to the solution.
c On exit x is the current approximation.
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,
c 3 if x(i) has only an upper bound.
c On exit nbd is unchanged.
c
c f is a double precision variable.
c On first entry f is unspecified.
c On final exit f is the value of the function at x.
c
c g is a double precision array of dimension n.
c On first entry g is unspecified.
c On final exit g is the value of the gradient at x.
c
c factr is a double precision variable.
c On entry factr >= 0 is specified by the user. The iteration
c 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.
c On exit factr is unchanged.
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 On exit pgtol is unchanged.
c
c ws, wy, sy, and wt are double precision working arrays used to
c store the following information defining the limited memory
c BFGS matrix:
c ws, of dimension n x m, stores S, the matrix of s-vectors;
c wy, of dimension n x m, stores Y, the matrix of y-vectors;
c sy, of dimension m x m, stores S'Y;
c ss, of dimension m x m, stores S'S;
c yy, of dimension m x m, stores Y'Y;
c wt, of dimension m x m, stores the Cholesky factorization
c of (theta*S'S+LD^(-1)L'); see eq.
c (2.26) in [3].
c
c wn is a double precision working array of dimension 2m x 2m
c used to store the LEL^T factorization of the indefinite matrix
c K = [-D -Y'ZZ'Y/theta L_a'-R_z' ]
c [L_a -R_z theta*S'AA'S ]
c
c where E = [-I 0]
c [ 0 I]
c
c snd is a double precision working array of dimension 2m x 2m
c used to store the lower triangular part of
c N = [Y' ZZ'Y L_a'+R_z']
c [L_a +R_z S'AA'S ]
c
c z(n),r(n),d(n),t(n), xp(n),wa(8*m) are double precision working arrays.
c z is used at different times to store the Cauchy point and
c the Newton point.
c xp is used to safeguard the projected Newton direction
c
c sg(m),sgo(m),yg(m),ygo(m) are double precision working arrays.
c
c index is an integer working array of dimension n.
c In subroutine freev, index is used to store the free and fixed
c variables at the Generalized Cauchy Point (GCP).
c
c iwhere is an integer working array of dimension n used to record
c the status of the vector x for GCP computation.
c iwhere(i)=0 or -3 if x(i) is free and has bounds,
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., no bounds on it.
c
c indx2 is an integer working array of dimension n.
c Within subroutine cauchy, indx2 corresponds to the array iorder.
c In subroutine freev, a list of variables entering and leaving
c the free set is stored in indx2, and it is passed on to
c subroutine formk with this information.
c
c task is a working string of characters of length 60 indicating
c the current job when entering and leaving this subroutine.
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 csave is a working string of characters of length 60.
c
c lsave is a logical working array of dimension 4.
c
c isave is an integer working array of dimension 23.
c
c dsave is a double precision working array of dimension 29.
c
c
c Subprograms called
c
c L-BFGS-B Library ... cauchy, subsm, lnsrlb, formk,
c
c errclb, prn1lb, prn2lb, prn3lb, active, projgr,
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