***********************************************************************
* *
* PSEN - A BUNDLE VARIABLE METRIC ALGORITHM FOR OPTIMIZATION OF *
* LARGE-SCALE NONSMOOTH PARTIALLY SEPARABLE FUNCTIONS. *
* *
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1. Introduction:
----------------
The double-precision FORTRAN 77 basic subroutine PSEN is designed
to find a close approximation to a local minimum of a partially separable
objective function
F(X) = FA_1(X) + FA_2(X) + ... + FA_NA(X).
Here X is a vector of NF variables and FA_I(X), 1 <= I <= NA, are
locally Lipschitz nonsmooth functions. We assume that NF and NA are
large, but partial functions FA_I(X), 1 <= I <= NA, depend on a small
number of variables. This implies that the nonsmooth mapping
AF(X) = [FA_1(X), FA_2(X), ..., FA_NA(X)] has a sparse subdifferential
containing generalized Jacobian matrices, which will be denoted by AG(X)
(they have NA rows and NF columns). The sparsity pattern of the Jacobian
matrix is stored in the coordinate form if ISPAS=1 or in the standard
compressed row format if ISPAS=2 using arrays IAG and JAG. For example,
if the Jacobian matrix has the following pattern
AG = | * * 0 * |
| * * * 0 |
| * 0 0 * |
| 0 * * 0 |
| * 0 * 0 |
(asterisks denote nonzero elements) then arrays IAG and JAG contain
elements
IAG(1)=1, IAG(2)=1, IAG(3)=1, IAG(4)=2, IAG(5)=2, IAG(6)=2,
IAG(7)=3, IAG(8)=3, IAG(9)=4, IAG(10)=4, IAG(11)=5, IAG(12)=5,
JAG(1)=1, JAG(2)=2, JAG(3)=4, JAG(4)=1, JAG(5)=2, JAG(6)=3,
JAG(7)=1, JAG(8)=4, JAG(9)=2, JAG(10)=3, JAG(11)=1, JAG(12)=3
if ISPAS=1 or
IAG(1)=1, IAG(2)=4, IAG(3)=7, IAG(4)=9, IAG(5)=11, IAG(6)=13,
JAG(1)=1, JAG(2)=2, JAG(3)=4, JAG(4)=1, JAG(5)=2, JAG(6)=3,
JAG(7)=1, JAG(8)=4, JAG(9)=2, JAG(10)=3, JAG(11)=1, JAG(12)=3
if ISPAS=2. In the first case, nonzero elements can be sorted in an
arbitrary order (not only by rows as in the above example). Arrays
IAG and JAG have to be declared with lengths NA+MA and MA at least,
respectively, where MA is the number of nonzero elements. In the
second case, nonzero elements can be sorted only by rows. Components
of IAG contain total numbers of nonzero elements in all previous
rows increased by 1 and elements of JAG contain corresponding column
indices (note that IAG has NA+1 elements and the last element is
equal to MA+1). Arrays IAG and JAG have to be declared with length
NA+1 and MA at least, respectively.
To simplify user's work, an additional easy to use subroutine
PSENU is added. It calls the basic general subroutine PSEN. All
subroutines contain a description of formal parameters and extensive
comments. Furthermore, test program TSENU is included, which contains
several test problems (see e.g. [2]). This test program serves as an
example for using the subroutine PSENU, verifies its correctness and
demonstrates its 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. Subroutine PSENU:
--------------------
The calling sequence is
CALL PSENU(NF,NA,MA,X,AF,IAG,JAG,IPAR,RPAR,F,GMAX,ISPAS,IPRNT,ITERM)
The arguments have the following meaning.
Argument Type Significance
----------------------------------------------------------------------
NF I Positive INTEGER variable that specifies the number of
variables of the partially separable function.
NA I Positive INTEGER variable that specifies the number of
partial functions.
MA I Number of nonzero elements in the Jacobian matrix. This
parameter is used as input only if ISPAS=1 (it defines
dimensions of arrays IAG and JAG in this case).
X(NF) U On input, DOUBLE PRECISION vector with the initial
estimate to the solution. On output, the approximation
to the minimum.
AF(NA) O DOUBLE PRECISION vector which contains values of partial
functions.
IAG(NA+1) I INTEGER array which contains pointers of the first
elements in rows of the Jacobian matrix.
JAG(MA) I INTEGER array which contains column indices of the
nonzero elements.
IPAR(7) U INTEGER parameters:
IPAR(1)=MIT, IPAR(2)=MFV, IPAR(3)-unused,
IPAR(4)=IEST, IPAR(5)-unused, IPAR(6)=MB,
IPAR(7)=IFIL.
Parameters MIT, MFV, IEST, MB are described in Section 3
together with other parameters of the subroutine PSEN.
Parameter IFIL specifies a relative size of the space
reserved for fill-in. The choice IFIL=0 causes that the
default value IFIL=1 will be taken.
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(8)=ETA3, RPAR(9)=ETA5.
Parameters XMAX, TOLX, TOLF, TOLB, TOLG, FMIN, ETA3, ETA5
are described in Section 3 together with other parameters
of the subroutine PSEN.
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 objective function.
ISPAS I INTEGER variable that specifies sparse structure of the
Jacobian matrix:
ISPAS= 1 - the coordinate form is used,
ISPAS= 2 - the standard row compresed format is used.
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. Values ITERM<=-40
detect a lack of space. In this case,
parameter IPAR(7)=IFIL has to be increased
(IFIL=2, IFIL=3, etc.).
The subroutine PSENU requires the user supplied subroutines FUN and
DFUN that define partial functions and their subgradients and have the
form
SUBROUTINE FUN(NF,KA,X,FA)
SUBROUTINE DFUN(NF,KA,X,GA)
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.
KA I INTEGER index of the partial function.
X(NF) I DOUBLE PRECISION an estimate to the solution.
FA O DOUBLE PRECISION value of the KA-th partial function at
the point X.
GA(NF) O DOUBLE PRECISION an arbitrary subgradient of the KA-th
partial function at the point X. Note that only nonzero
elements of this subgradient have to be assigned.
3. Subroutine PSEN:
-------------------
This general subroutine is called from all subroutines described
in Section 2. The calling sequence is
CALL PSEN(NF,NA,MB,MMAX,X,IX,AF,AG,AGO,AH,GA,G,H,IH,JH,IAG,
& JAG,S,XO,GO,XS,GS,GP,AX,AY,AZ,PSL,PERM,INVP,WN11,WN12,WN13,
& WN14,XMAX,TOLX,TOLF,TOLB,TOLG,FMIN,ETA3,ETA5,GMAX,F,MIT,MFV,
& IEST,IPRNT,ITERM)
The arguments NF, NA, X, AF, IAG, JAG, GMAX, F, IPRNT, ITERM have the
same meaning as in Section 2. Other arguments have the following meaning:
Argument Type Significance
----------------------------------------------------------------------
MB I INTEGER dimension of a bundle used in the line search.
MMAX I INTEGER size of array H.
IX(NF) A INTEGER auxiliary array.
AG(MA) A DOUBLE PRECISION nonzero elements of the Jacobian
matrix.
AGO(MA) A DOUBLE PRECISION auxiliary array.
AH(MH) A DOUBLE PRECISION approximation of the partitioned
Hessian matrix.
GA(NF) A DOUBLE PRECISION gradient of the partial function.
G(NF) A DOUBLE PRECISION gradient of the objective function.
H(MMAX) A DOUBLE PRECISION nonzero elements of the approximation
of the Hessian matrix and nonzero elements of the
Choleski factor.
IH(NF+1) I INTEGER array which contains pointers of the diagonal
elements in the upper part of the Hessian matrix.
JH(MMAX) I INTEGER array which contains column indices of the
nonzero elements and additional working space for the
Choleski factor.
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.
XS(NF) A DOUBLE PRECISION auxiliary array.
GS(NF) A DOUBLE PRECISION auxiliary array.
GP(NF) A DOUBLE PRECISION auxiliary array.
AX(NF*MB) A DOUBLE PRECISION auxiliary array.
AY(NF*MB) A DOUBLE PRECISION auxiliary array.
AZ(4*MB) A DOUBLE PRECISION auxiliary array.
PSL(NF+1) A INTEGER pointer vector in the compact form of the
Choleski factor.
PERM(NF) A INTEGER permutation vector.
INVP(NF) A INTEGER inverse permutation vector.
WN11(NF+1) A INTEGER auxiliary array.
WN12(NF+1) A INTEGER auxiliary array.
WN13(NF+1) A INTEGER auxiliary array.
WN14(NF+1) A INTEGER auxiliary array.
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-12 will be taken.
TOLB U DOUBLE PRECISION minimum acceptable function value;
the choice TOLB=0 causes that the default value
TOLB=FMIN+1.0D-12 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-8 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.
ETA3 U DOUBLE PRECISION correction parameter; the choice ETA3=0
causes that the default value ETA3=1.0D-12 will be taken.
ETA5 U DOUBLE PRECISION parameter for subgradient locality
measure; the choice ETA5=0 causes that the default value
ETA5=1.0D-12 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.
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. A suitable choice of parameter ETA5
can sometimes improve the efficiency of the method.
The subroutine PSEN requires the user supplied subroutines FUN
and DFUN which are described in Section 2.
4. Verification of the subroutines:
-----------------------------------
Subroutine PSENU can be verified and tested using the program
TSENU. This program calls the subroutines TIUB15 (initiation), TAFU15
(function evaluation) and TAGU15 (gradient evaluation) containing
22 unconstrained test problems with at most 200 variables [2]. The
results obtained by the program TSENU on a PC computer with Microsoft
Power Station Fortran compiler have the following form.
NIT= 3124 NFV= 3134 NFG= 3134 F= 0.287703261E-08 G= 0.582E-08 ITERM= 4
NIT= 286 NFV= 287 NFG= 287 F= 0.379499216E-08 G= 0.203E-06 ITERM= 2
NIT= 71 NFV= 71 NFG= 71 F= 0.233196848E-09 G= 0.100E-07 ITERM= 4
NIT= 40 NFV= 40 NFG= 40 F= 126.863549 G= 0.699E-08 ITERM= 4
NIT= 282 NFV= 282 NFG= 282 F= 0.732927514E-07 G= 0.400E-08 ITERM= 4
NIT= 344 NFV= 344 NFG= 344 F= 0.836329152E-08 G= 0.326E-08 ITERM= 4
NIT= 286 NFV= 287 NFG= 287 F= 2391.16999 G= 0.673E-04 ITERM= 2
NIT= 610 NFV= 611 NFG= 611 F= 0.317244739E-05 G= 0.548E-08 ITERM= 4
NIT= 2514 NFV= 2516 NFG= 2516 F= 552.380551 G= 0.448E-08 ITERM= 4
NIT= 907 NFV= 907 NFG= 907 F= 131.888476 G= 0.579E-08 ITERM= 4
NIT= 269 NFV= 271 NFG= 271 F= 0.173668302E-09 G= 0.266E-08 ITERM= 4
NIT= 1805 NFV= 1810 NFG= 1810 F= 621.128947 G= 0.906E-02 ITERM= 2
NIT= 680 NFV= 681 NFG= 681 F= 2940.50943 G= 0.140E-03 ITERM= 2
NIT= 370 NFV= 370 NFG= 370 F= 112.314954 G= 0.622E-08 ITERM= 4
NIT= 364 NFV= 364 NFG= 364 F= 36.0935676 G= 0.986E-08 ITERM= 4
NIT= 1004 NFV= 1004 NFG= 1004 F= 13.2000000 G= 0.904E-08 ITERM= 4
NIT= 380 NFV= 380 NFG= 380 F= 0.268534232E-01 G= 0.871E-09 ITERM= 4
NIT=15319 NFV=15321 NFG=15321 F= 0.589970806E-08 G= 0.925E-08 ITERM= 4
NIT= 3972 NFV= 4056 NFG= 4056 F= 0.565862690E-08 G= 0.887E-08 ITERM= 4
NIT= 774 NFV= 988 NFG= 988 F= 0.406495193E-08 G= 0.468E-08 ITERM= 4
NIT= 247 NFV= 248 NFG= 248 F= 264.000000 G= 0.364E-03 ITERM= 2
NIT= 1191 NFV= 1192 NFG= 1192 F= 593.360762 G= 0.145E-03 ITERM= 2
NITER =34839 NFVAL =35164 NSUCC = 22
TIME= 0:00:13.49
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.
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.