***********************************************************************
* *
* PLIS - A LIMITED MEMORY VARIABLE METRIC ALGORITHM FOR *
* LARGE-SCALE OPTIMIZATION. *
* *
***********************************************************************
1. Introduction:
----------------
The double-precision FORTRAN 77 basic subroutine PLIS 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 PLIS:
PLISU - unconstrained large-scale optimization,
PLISS - large-scale optimization with simple bounds.
All subroutines contain a description of formal parameters and
extensive comments. Furthermore, two test programs TLISU and TLISS 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). Note that the arguments of the type I can be changed on
output under some circumstances, especially if improper input values
were given. 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 specifies the number of
gradient evaluations.
NFH O Positive INTEGER variable that specifies the number of
Hessian evaluations.
2. Subroutines PLISU, PLISS:
----------------------------
The calling sequences are
CALL PLISU(NF,X,IPAR,RPAR,F,GMAX,IPRNT,ITERM)
CALL PLISS(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 if NB>0) 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 if NB>0).
XU(NF) I DOUBLE PRECISION vector with upper bounds for variables
(significant only if NB>0).
IPAR(7) I INTEGER parameters:
IPAR(1)=MIT, IPAR(2)=MFV, IPAR(3)-NONE,
IPAR(4)=IEST, IPAR(5)-NONE, IPAR(6)-NONE,
IPAR(7)=MF.
Parameters MIT, MFV, IEST, MF are described in Section 3
together with other parameters of the subroutine PLIS.
RPAR(9) I DOUBLE PRECISION parameters:
RPAR(1)=XMAX, RPAR(2)=TOLX, RPAR(3)=TOLF,
RPAR(4)=TOLB, RPAR(5)=TOLG, RPAR(6)=FMIN,
RPAR(7)-NONE, RPAR(6)-NONE, RPAR(9)-NONE.
Parameters XMAX, TOLX, TOLF, TOLB, TOLG, FMIN are
described in Section 3 together with other parameters
of the subroutine PLIS.
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 Lagrangian 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 PLISU, PLISS 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 PLIS:
-------------------
This general subroutine is called from all subroutines described
in Section 2. The calling sequence is
CALL PLIS(NF,NB,X,IX,XL,XU,GF,S,XO,GO,UO,VO,XMAX,TOLX,TOLF,TOLB,
& TOLG,FMIN,GMAX,F,MIT,MFV,IEST,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*MF) A DOUBLE PRECISION array which contains increments of
variables.
GO(NF*MF) A DOUBLE PRECISION array which contains increments of
gradients.
UO(MF) A DOUBLE PRECISION Auxiliary array.
VO(MF) A DOUBLE PRECISION Auxiliary array.
XMAX I DOUBLE PRECISION maximum stepsize; the choice XMAX=0
causes that the default value 1.0D+16 will be taken.
TOLX I 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 I 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 I DOUBLE PRECISION minimum acceptable function value;
the choice TOLB=0 causes that the default value
TOLB=FMIN+1.0D-16 will be taken.
TOLG I 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 I DOUBLE PRECISION lower bound for the minimum function
value.
MIT I INTEGER variable that specifies the maximum number of
iterations; the choice MIT=0 causes that the default
value 9000 will be taken.
MFV I 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.
MF I The number of limited-memory variable metric updates
in each iteration (they use 2*MF stored vectors).
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 PLIS requires the user supplied subroutines OBJ
and DOBJ which are described in Section 2.
4. Verification of the subroutines:
-----------------------------------
Subroutine PLISU can be verified and tested using the program
TLISU. 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 TLISU on a PC computer with Microsoft
Power Station Fortran compiler have the following form.
NIT= 4988 NFV= 5554 NFG= 5554 F= 0.963780013E-14 G= 0.891E-06 ITERM= 4
NIT= 425 NFV= 454 NFG= 454 F= 14.9944763 G= 0.773E-05 ITERM= 2
NIT= 74 NFV= 78 NFG= 78 F= 0.655101686E-09 G= 0.539E-06 ITERM= 4
NIT= 103 NFV= 112 NFG= 112 F= 269.499543 G= 0.899E-06 ITERM= 4
NIT= 24 NFV= 26 NFG= 26 F= 0.130639280E-11 G= 0.671E-06 ITERM= 4
NIT= 30 NFV= 31 NFG= 31 F= 0.216102227E-10 G= 0.946E-06 ITERM= 4
NIT= 38 NFV= 43 NFG= 43 F= 335.137433 G= 0.730E-06 ITERM= 4
NIT= 29 NFV= 33 NFG= 33 F= 761774.954 G= 0.432E-03 ITERM= 2
NIT= 13 NFV= 16 NFG= 16 F= 316.436141 G= 0.369E-06 ITERM= 4
NIT= 1540 NFV= 1582 NFG= 1582 F= -124.630000 G= 0.124E-04 ITERM= 2
NIT= 114 NFV= 138 NFG= 138 F= 10.7765879 G= 0.380E-06 ITERM= 4
NIT= 248 NFV= 267 NFG= 267 F= 982.273617 G= 0.123E-04 ITERM= 2
NIT= 7 NFV= 8 NFG= 8 F= 0.165734137E-12 G= 0.453E-06 ITERM= 4
NIT= 10 NFV= 12 NFG= 12 F= 0.128729169E-08 G= 0.916E-06 ITERM= 4
NIT= 2830 NFV= 2929 NFG= 2929 F= 1.92401599 G= 0.936E-06 ITERM= 4
NIT= 196 NFV= 210 NFG= 210 F= -427.404476 G= 0.991E-05 ITERM= 2
NIT= 1007 NFV= 1032 NFG= 1032 F=-0.379921091E-01 G= 0.876E-06 ITERM= 4
NIT= 1449 NFV= 1474 NFG= 1474 F=-0.245741193E-01 G= 0.862E-06 ITERM= 4
NIT= 1393 NFV= 1431 NFG= 1431 F= 59.5986241 G= 0.259E-05 ITERM= 2
NIT= 2129 NFV= 2191 NFG= 2191 F= -1.00013520 G= 0.908E-06 ITERM= 4
NIT= 2120 NFV= 2169 NFG= 2169 F= 2.13866377 G= 0.927E-06 ITERM= 4
NIT= 1305 NFV= 1346 NFG= 1346 F= 1.00000000 G= 0.982E-06 ITERM= 4
NITER =20072 NFVAL =21136 NSUCC = 22
TIME= 0:00:10.78
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 PLISS can be verified and tested using the program
TLISS. 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 TLISS on a PC computer with Microsoft Power
Station Fortran compiler have the following form.
NIT= 5063 NFV= 5738 NFG= 5738 F= 0.00000000 G= 0.000E+00 ITERM= 3
NIT= 3167 NFV= 4664 NFG= 4664 F= 3926.45961 G= 0.626E-04 ITERM= 2
NIT= 113 NFV= 124 NFG= 124 F= 0.459503394E-12 G= 0.600E-06 ITERM= 4
NIT= 59 NFV= 64 NFG= 64 F= 269.522686 G= 0.838E-06 ITERM= 4
NIT= 24 NFV= 26 NFG= 26 F= 0.130639280E-11 G= 0.671E-06 ITERM= 4
NIT= 30 NFV= 31 NFG= 31 F= 0.216102227E-10 G= 0.946E-06 ITERM= 4
NIT= 33 NFV= 40 NFG= 40 F= 337.722479 G= 0.592E-06 ITERM= 4
NIT= 50 NFV= 55 NFG= 55 F= 761925.725 G= 0.240E-03 ITERM= 2
NIT= 505 NFV= 508 NFG= 508 F= 428.056916 G= 0.334E-07 ITERM= 4
NIT= 1167 NFV= 1227 NFG= 1227 F= -81.0913589 G= 0.100E-04 ITERM= 2
NIT= 20 NFV= 26 NFG= 26 F= 96517.2947 G= 0.745E-05 ITERM= 2
NIT= 91 NFV= 109 NFG= 109 F= 4994.21410 G= 0.104E-04 ITERM= 2
NIT= 7 NFV= 8 NFG= 8 F= 0.165734137E-12 G= 0.453E-06 ITERM= 4
NIT= 10 NFV= 12 NFG= 12 F= 0.128729169E-08 G= 0.916E-06 ITERM= 4
NIT= 2830 NFV= 2929 NFG= 2929 F= 1.92401599 G= 0.936E-06 ITERM= 4
NIT= 178 NFV= 184 NFG= 184 F= -427.391653 G= 0.107E-04 ITERM= 2
NIT= 1007 NFV= 1032 NFG= 1032 F=-0.379921091E-01 G= 0.876E-06 ITERM= 4
NIT= 1449 NFV= 1474 NFG= 1474 F=-0.245741193E-01 G= 0.862E-06 ITERM= 4
NIT= 1561 NFV= 1595 NFG= 1595 F= 1654.94525 G= 0.112E-04 ITERM= 2
NIT= 2075 NFV= 2121 NFG= 2121 F= -1.00013520 G= 0.916E-06 ITERM= 4
NIT= 1361 NFV= 1389 NFG= 1389 F= 2.41354873 G= 0.709E-06 ITERM= 4
NIT= 1562 NFV= 1598 NFG= 1598 F= 1.00000000 G= 0.786E-06 ITERM= 4
NITER =22362 NFVAL =24954 NSUCC = 22
TIME= 0:00:12.39
References:
-----------
[1] Luksan L., Matonoha C., Vlcek J.: LSA: Algorithms for large-scale
unconstrained and box constrained optimization Technical Report V-896.
Prague, ICS AS CR, 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.