***********************************************************************
* *
* PLIP - A LIMITED MEMORY SHIFTED VARIABLE METRIC ALGORITHM *
* FOR LARGE-SCALE OPTIMIZATION. *
* *
***********************************************************************
1. Introduction:
----------------
The double-precision FORTRAN 77 basic subroutine PLIP is designed
to find a close approximation to a local minimum of a nonlinear
function F(X) with simple bounds on variables. Here X is a vector of NF
variables and F(X) is a smooth function. We suppose that NF is large
but the sparsity pattern of the Hessian matrix is not known (or the
Hessian matrix is dense). Simple bounds are assumed in the form
X(I) unbounded if IX(I) = 0,
XL(I) <= X(I) if IX(I) = 1,
X(I) <= XU(I) if IX(I) = 2,
XL(I) <= X(I) <= XU(I) if IX(I) = 3,
XL(I) = X(I) = XU(I) if IX(I) = 5,
where 1 <= I <= NF. To simplify user's work, two additional easy to use
subroutines are added. They call the basic general subroutine PLIP:
PLIPU - unconstrained large-scale optimization,
PLIPS - large-scale optimization with simple bounds.
All subroutines contain a description of formal parameters and
extensive comments. Furthermore, two test programs TLIPU and TLIPS are
included, which contain several test problems (see e.g. [2]). These
test programs serve as examples for using the subroutines, verify their
correctness and demonstrate their efficiency.
In this short guide, we describe all subroutines which can be
called from the user's program. A detailed description of the method is
given in [1]. In the description of formal parameters, we introduce a
type of the argument that specifies whether the argument must have a
value defined on entry to the subroutine (I), whether it is a value
which will be returned (O), or both (U), or whether it is an auxiliary
value (A). Besides formal parameters, we can use a COMMON /STAT/ block
containing statistical information. This block, used in each subroutine
has the following form:
COMMON /STAT/ NRES,NDEC,NIN,NIT,NFV,NFG,NFH
The arguments have the following meaning:
Argument Type Significance
----------------------------------------------------------------------
NRES O Positive INTEGER variable that indicates the number of
restarts.
NDEC O Positive INTEGER variable that indicates the number of
matrix decompositions.
NIN O Positive INTEGER variable that indicates the number of
inner iterations (for solving linear systems).
NIT O Positive INTEGER variable that indicates the number of
iterations.
NFV O Positive INTEGER variable that indicates the number of
function evaluations.
NFG O Positive INTEGER variable that indicates the number of
gradient evaluations.
NFH O Positive INTEGER variable that indicates the number of
Hessian evaluations.
2. Subroutines PLIPU, PLIPS:
----------------------------
The calling sequences are
CALL PLIPU(NF,X,IPAR,RPAR,F,GMAX,IPRNT,ITERM)
CALL PLIPS(NF,X,IX,XL,XU,IPAR,RPAR,F,GMAX,IPRNT,ITERM)
The arguments have the following meaning.
Argument Type Significance
----------------------------------------------------------------------
NF I Positive INTEGER variable that specifies the number of
variables of the objective function.
X(NF) U On input, DOUBLE PRECISION vector with the initial
estimate to the solution. On output, the approximation
to the minimum.
IX(NF) I On input (significant only for PLIPS) INTEGER vector
containing the simple bounds types:
IX(I)=0 - the variable X(I) is unbounded,
IX(I)=1 - the lower bound X(I) >= XL(I),
IX(I)=2 - the upper bound X(I) <= XU(I),
IX(I)=3 - the two side bound XL(I) <= X(I) <= XU(I),
IX(I)=5 - the variable X(I) is fixed (given by its
initial estimate).
XL(NF) I DOUBLE PRECISION vector with lower bounds for variables
(significant only for PLIPS).
XU(NF) I DOUBLE PRECISION vector with upper bounds for variables
(significant only for PLIPS).
IPAR(7) U INTEGER parameters:
IPAR(1)=MIT, IPAR(2)=MFV, IPAR(3)-unused,
IPAR(4)=IEST, IPAR(5)-MET, IPAR(6)-unused,
IPAR(7)=MF.
Parameters MIT, MFV, IEST, MET, MF are described
in Section 3 together with other parameters of the
subroutine PLIP.
RPAR(9) U DOUBLE PRECISION parameters:
RPAR(1)=XMAX , RPAR(2)=TOLX, RPAR(3)=TOLF,
RPAR(4)=TOLB, RPAR(5)=TOLG, RPAR(6)=FMIN,
RPAR(7)-unused, RPAR(6)-unused, RPAR(9)-unused.
Parameters XMAX, TOLX, TOLF, TOLB, TOLG, FMIN are
described in Section 3 together with other parameters
of the subroutine PLIP.
F O DOUBLE PRECISION value of the objective function at the
solution X.
GMAX O DOUBLE PRECISION maximum absolute value of a partial
derivative of the objetive function.
IPRNT I INTEGER variable that specifies PRINT:
IPRNT= 0 - print is suppressed,
IPRNT= 1 - basic print of final results,
IPRNT=-1 - extended print of final results,
IPRNT= 2 - basic print of intermediate and final
results,
IPRNT=-2 - extended print of intermediate and final
results.
ITERM O INTEGER variable that indicates the cause of termination:
ITERM= 1 - if |X - XO| was less than or equal to TOLX
in two subsequent iterations,
ITERM= 2 - if |F - FO| was less than or equal to TOLF
in two subsequent iterations,
ITERM= 3 - if F is less than or equal to TOLB,
ITERM= 4 - if GMAX is less than or equal to TOLG,
ITERM= 6 - if termination criterion was not satisfied,
but the solution is probably acceptable,
ITERM=11 - if NIT exceeded MIT,
ITERM=12 - if NFV exceeded MFV,
ITERM< 0 - if the method failed.
The subroutines PLIPU, PLIPS require the user supplied subroutines
OBJ and DOBJ that define the objective function and its gradient and
have the form
SUBROUTINE OBJ(NF,X,F)
SUBROUTINE DOBJ(NF,X,G)
The arguments of the user supplied subroutines have the following
meaning.
Argument Type Significance
----------------------------------------------------------------------
NF I Positive INTEGER variable that specifies the number of
variables of the objective function.
X(NF) I DOUBLE PRECISION an estimate to the solution.
F O DOUBLE PRECISION value of the objective function at the
point X.
G(NF) O DOUBLE PRECISION gradient of the objective function
at the point X.
3. Subroutine PLIP:
-------------------
This general subroutine is called from all subroutines described
in Section 2. The calling sequence is
CALL PLIP(NF,NB,X,IX,XL,XU,GF,S,XO,GO,SO,XM,XR,GR,XMAX,TOLX,TOLF,
& TOLB,TOLG,FMIN,GMAX,F,MIT,MFV,IEST,MET,MF,IPRNT,ITERM)
The arguments NF, NB, X, IX, XL, XU, GMAX, F, IPRNT, ITERM, have the
same meaning as in Section 2. Other arguments have the following meaning:
Argument Type Significance
----------------------------------------------------------------------
GF(NF) A DOUBLE PRECISION gradient of the objective function.
S(NF) A DOUBLE PRECISION direction vector.
XO(NF) A DOUBLE PRECISION array which contains increments of
variables.
GO(NF) A DOUBLE PRECISION array which contains increments of
gradients.
SO(NF) A DOUBLE PRECISION auxiliary array.
XM(NF*MF) A DOUBLE PRECISION array which contains columns
of the updated matrix stored in the product form.
XR(MF) A DOUBLE PRECISION array which contains reduced
increments of variables.
GR(MF) A DOUBLE PRECISION array which contains reduced
increments of gradients.
XMAX U DOUBLE PRECISION maximum stepsize; the choice XMAX=0
causes that the default value 1.0D+16 will be taken.
TOLX U DOUBLE PRECISION tolerance for the change of the
coordinate vector X; the choice TOLX=0 causes that the
default value TOLX=1.0D-16 will be taken.
TOLF U DOUBLE PRECISION tolerance for the change of function
values; the choice TOLF=0 causes that the default
value TOLF=1.0D-14 will be taken.
TOLB U DOUBLE PRECISION minimum acceptable function value;
the choice TOLB=0 causes that the default value
TOLB=FMIN+1.0D-16 will be taken.
TOLG U DOUBLE PRECISION tolerance for the Lagrangian function
gradient; the choice TOLG=0 causes that the default
value TOLG=1.0D-6 will be taken.
FMIN U DOUBLE PRECISION lower bound for the minimum function
value. It is significant only if IEST=1. If IEST=0,
the default value FMIN=-1.0D+60 will be taken.
MIT U INTEGER variable that specifies the maximum number of
iterations; the choice MIT=0 causes that the default
value 9000 will be taken.
MFV U INTEGER variable that specifies the maximum number of
function evaluations; the choice MFV=0 causes that
the default value 9000 will be taken.
IEST I INTEGER estimation of the minimum functiom value for
the line search:
IEST=0 - estimation is not used,
IEST=1 - lower bound FMIN is used as an estimation
for the minimum function value.
MET U INTEGER variable that specifies the limited-memory
method:
MET=1 - rank-one method,
MET=2 - rank-two method.
The choice MET=0 causes that the default value MET=2
will be taken.
MF U The number of limited-memory variable metric updates
in each iteration (they use MF stored vectors).
The choice MF=0 causes that the default value MF=10
will be taken.
The choice of parameter XMAX can be sensitive in many cases. First, the
objective function can be evaluated only in a relatively small region
(if it contains exponentials) so that the maximum stepsize is necessary.
Secondly, the problem can be very ill-conditioned far from the solution
point so that large steps can be unsuitable. Finally, if the problem has
more local solutions, a suitably chosen maximum stepsize can lead to
obtaining a better local solution.
The subroutine PLIP requires the user supplied subroutines OBJ
and DOBJ which are described in Section 2.
4. Verification of the subroutines:
-----------------------------------
Subroutine PLIPU can be verified and tested using the program
TLIPU. This program calls the subroutines TIUD14 (initiation), TFFU14
(function evaluation) and TFGU14 (gradient evaluation) containing
22 unconstrained test problems with at most 1000 variables [2]. The
results obtained by the program TLIPU on a PC computer with Microsoft
Power Station Fortran compiler have the following form.
NIT= 5383 NFV= 5417 NFG= 5417 F= 0.601022658E-13 G= 0.599E-06 ITERM= 4
NIT= 530 NFV= 557 NFG= 557 F= 3.57276719 G= 0.124E-05 ITERM= 2
NIT= 125 NFV= 128 NFG= 128 F= 0.338270284E-12 G= 0.518E-06 ITERM= 4
NIT= 109 NFV= 114 NFG= 114 F= 269.499543 G= 0.669E-06 ITERM= 4
NIT= 26 NFV= 27 NFG= 27 F= 0.710072396E-11 G= 0.951E-06 ITERM= 4
NIT= 35 NFV= 36 NFG= 36 F= 0.142942272E-10 G= 0.737E-06 ITERM= 4
NIT= 36 NFV= 41 NFG= 41 F= 336.937181 G= 0.956E-06 ITERM= 4
NIT= 33 NFV= 36 NFG= 36 F= 761774.954 G= 0.192E-02 ITERM= 2
NIT= 15 NFV= 18 NFG= 18 F= 316.436141 G= 0.264E-06 ITERM= 4
NIT= 2003 NFV= 2030 NFG= 2030 F= -124.950000 G= 0.116E-04 ITERM= 2
NIT= 157 NFV= 175 NFG= 175 F= 10.7765879 G= 0.299E-06 ITERM= 4
NIT= 337 NFV= 350 NFG= 350 F= 982.273617 G= 0.145E-04 ITERM= 2
NIT= 9 NFV= 10 NFG= 10 F= 0.230414406E-14 G= 0.642E-07 ITERM= 4
NIT= 8 NFV= 10 NFG= 10 F= 0.128834241E-08 G= 0.977E-06 ITERM= 4
NIT= 1226 NFV= 1256 NFG= 1256 F= 1.92401599 G= 0.970E-06 ITERM= 4
NIT= 237 NFV= 246 NFG= 246 F= -427.404476 G= 0.501E-04 ITERM= 2
NIT= 598 NFV= 604 NFG= 604 F=-0.379921091E-01 G= 0.908E-06 ITERM= 4
NIT= 989 NFV= 998 NFG= 998 F=-0.245741193E-01 G= 0.975E-06 ITERM= 4
NIT= 1261 NFV= 1272 NFG= 1272 F= 59.5986241 G= 0.410E-05 ITERM= 2
NIT= 2045 NFV= 2058 NFG= 2058 F= -1.00013520 G= 0.911E-06 ITERM= 4
NIT= 2175 NFV= 2196 NFG= 2196 F= 2.13866377 G= 0.996E-06 ITERM= 4
NIT= 1261 NFV= 1292 NFG= 1292 F= 1.00000000 G= 0.927E-06 ITERM= 4
NITER =18598 NFVAL =18871 NSUCC = 22
TIME= 0:00:10.63
The rows corresponding to individual test problems contain the number of
iterations NIT, the number of function evaluations NFV, the number of
gradient evaluations NFG, the final value of the objective function F,
the norm of gradient G and the cause of termination ITERM.
Subroutine PLIPS can be verified and tested using the program
TLIPS. This program calls the subroutines TIUD14 (initiation), TFFU14
(function evaluation), TFGU14 (gradient evaluation) containing 22 box
constrained test problems with at most 1000 variables [2]. The results
obtained by the program TLIPS on a PC computer with Microsoft Power
Station Fortran compiler have the following form.
NIT= 5263 NFV= 5321 NFG= 5321 F= 0.530131995E-13 G= 0.370E-05 ITERM= 2
NIT= 2293 NFV= 2447 NFG= 2447 F= 3930.43962 G= 0.251E-04 ITERM= 2
NIT= 127 NFV= 132 NFG= 132 F= 0.210550150E-12 G= 0.437E-06 ITERM= 4
NIT= 70 NFV= 72 NFG= 72 F= 269.522686 G= 0.794E-06 ITERM= 4
NIT= 26 NFV= 27 NFG= 27 F= 0.710072396E-11 G= 0.951E-06 ITERM= 4
NIT= 35 NFV= 36 NFG= 36 F= 0.142942272E-10 G= 0.737E-06 ITERM= 4
NIT= 37 NFV= 43 NFG= 43 F= 336.937181 G= 0.133E-05 ITERM= 2
NIT= 59 NFV= 65 NFG= 65 F= 761925.725 G= 0.399E-03 ITERM= 2
NIT= 508 NFV= 510 NFG= 510 F= 428.056916 G= 0.776E-06 ITERM= 4
NIT= 1253 NFV= 1277 NFG= 1277 F= -82.5400568 G= 0.120E-04 ITERM= 2
NIT= 13 NFV= 19 NFG= 19 F= 96517.2947 G= 0.150E-04 ITERM= 2
NIT= 95 NFV= 102 NFG= 102 F= 4994.21410 G= 0.790E-04 ITERM= 2
NIT= 9 NFV= 10 NFG= 10 F= 0.230414406E-14 G= 0.642E-07 ITERM= 4
NIT= 8 NFV= 10 NFG= 10 F= 0.128834241E-08 G= 0.977E-06 ITERM= 4
NIT= 1226 NFV= 1256 NFG= 1256 F= 1.92401599 G= 0.970E-06 ITERM= 4
NIT= 227 NFV= 228 NFG= 228 F= -427.391653 G= 0.952E-05 ITERM= 2
NIT= 598 NFV= 604 NFG= 604 F=-0.379921091E-01 G= 0.908E-06 ITERM= 4
NIT= 989 NFV= 998 NFG= 998 F=-0.245741193E-01 G= 0.975E-06 ITERM= 4
NIT= 1367 NFV= 1383 NFG= 1383 F= 1654.94525 G= 0.105E-04 ITERM= 2
NIT= 2274 NFV= 2303 NFG= 2303 F= -1.00013520 G= 0.798E-06 ITERM= 4
NIT= 1196 NFV= 1211 NFG= 1211 F= 2.41354873 G= 0.975E-06 ITERM= 4
NIT= 1361 NFV= 1381 NFG= 1381 F= 1.00000000 G= 0.962E-06 ITERM= 4
NITER =19034 NFVAL =19435 NSUCC = 22
TIME= 0:00:11.09
References:
-----------
[1] Luksan L., Matonoha C., Vlcek J.: LSA: Algorithms for large-scale
unconstrained and box constrained optimization. Research Report
V-896, Institute of Computer Science, Academy of Sciences of the
Czech Republic, Prague, Czech Republic, 2004.
[2] Luksan L., Vlcek J.: Sparse and partially separable test problems
for unconstrained and equality constrained optimization. Research
Report V-767, Institute of Computer Science, Academy of Sciences
of the Czech Republic, Prague, Czech Republic, 1998.