***********************************************************************
* *
* PNEC - A DISCRETE NEWTON METHOD WITH ITERATIVE CG-BASED *
* TRUST-REGION SUBALGORITHMS FOR LARGE-SCALE OPTIMIZATION *
* PROBLEMS WITH SPARSE HESSIAN MATRICES. *
* *
***********************************************************************
1. Introduction:
----------------
The double-precision FORTRAN 77 basic subroutine PNEC 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
and the sparsity pattern of the Hessian matrix is known. 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. The sparsity pattern of the Hessian matrix (only
the upper part) is stored in the coordinate form if ISPAS=1 or in the
standard compressed row format if ISPAS=2 using arrays IH and JH. For
example, if the Jacobian matrix has the following pattern
H = | * * * 0 * |
| * * 0 * 0 |
| * 0 * 0 * |
| 0 * 0 * 0 |
| * 0 * 0 * |
(asterisks denote nonzero elements) then arrays IH and JH contain
elements
IH(1)=1, IH(2)=1, IH(3)=1, IH(4)=1, IH(5)=2, IH(6)=2, IH(7)=3,
IH(8)=3, IH(9)=4, IH(10)=5,
JH(1)=1, JH(2)=2, JH(3)=3, JH(4)=5, JH(5)=2, JH(6)=4, JH(7)=3,
JH(8)=5, JH(9)=4, JH(10)=5
if ISPAS=1 or
IH(1)=1, IH(2)=5, IH(3)=7, IH(4)=9, IH(5)=10, IH(6)=11,
JH(1)=1, JH(2)=2, JH(3)=3, JH(4)=5, JH(5)=2, JH(6)=4, JH(7)=3,
JH(8)=5, JH(9)=4, JH(10)=5
if ISPAS=2. In the first case, nonzero elements in the upper part of
the Hessian matrix can be sorted in an arbitrary order (not only by
rows as in the above example) and arrays IH and JH have to be declared
with lengths NF+MH at least, where MH is the number of nonzero elements.
In the second case, nonzero elements can be sorted only by rows.
Components of IH contain addresses of the diagonal elements in this
sequence and components of JH contain corresponding column indices
(note that IH has NF+1 elements and the last element is equal to MH+1).
Arrays IH and JH have to be declared with lengths NF+1 and MH at least,
respectively.
To simplify user's work, two additional easy to use subroutines
are added. They call the basic general subroutine PNEC:
PNECU - unconstrained large-scale optimization,
PNECS - large-scale optimization with simple bounds.
All subroutines contain a description of formal parameters and
extensive comments. Furthermore, two test programs TNECU and TNECS 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 PNECU, PNECS:
----------------------------
The calling sequences are
CALL PNECU(NF,MH,X,IH,JH,IPAR,RPAR,F,GMAX,ISPAS,IPRNT,ITERM)
CALL PNECS(NF,MH,X,IX,XL,XU,IH,JH,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 objective function.
MH I Number of nonzero elements in the upper part of the
Hessian matrix. This parameter is used as input only if
ISPAS=1 (it defines dimensions of arrays IH and JH 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.
IX(NF) I On input (significant only for PNECS) 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 PNECS).
XU(NF) I DOUBLE PRECISION vector with upper bounds for variables
(significant only if NB>0).
IH(NF+1) I INTEGER array which contains pointers of the diagonal
elements in the upper part of the Hessian matrix.
JH(MH) I INTEGER array which contains column indices of the
nonzero elements and additional working space for the
Choleski factor.
IPAR(7) U INTEGER parameters:
IPAR(1)=MIT, IPAR(2)=MFV, IPAR(3)=MFG,
IPAR(4)=IEST, IPAR(5)=MOS1, IPAR(6)=MOS2,
IPAR(7)=IFIL.
Parameters MIT, MFV, MFG, IEST, MOS1, MOS2 are
described in Section 3 together with other parameters
of the subroutine PNEC. 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)=XDEL, RPAR(8)-unused, RPAR(9)-unused.
Parameters XMAX, TOLX, TOLF, TOLB, TOLG, FMIN, XDEL
are described in Section 3 together with other
parameters of the subroutine PNEC.
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
Hessian 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=13 - if NFG exceeded MFG,
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 subroutines PNECU, PNECS 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 PNEC:
-------------------
This general subroutine is called from all subroutines described
in Section 2. The calling sequence is
CALL PNEC(NF,NB,MMAX,X,IX,XL,XU,GF,HF,IH,JH,S,XO,GO,XS,GS,COL,
& WN11,WN12,IW,XMAX,TOLX,TOLF,TOLB,TOLG,FMIN,XDEL,GMAX,F,MIT,MFV,
& MFG,IEST,MOS1,MOS2,IPRNT,ITERM)
The arguments NF, NB, X, IX, XL, XU, IH, JH, GMAX, F, IPRNT, ITERM,
have the same meaning as in Section 2. Other arguments have the following
meaning:
Argument Type Significance
----------------------------------------------------------------------
MMAX I INTEGER size of array H.
GF(NF) A DOUBLE PRECISION gradient of the objective function.
HF(MMAX) A DOUBLE PRECISION nonzero elements of the original
Hessian matrix and nonzero elements of 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.
COL(NF) A INTEGER auxiliary array.
WN11(NF+1) A INTEGER auxiliary array.
WN12(NF+1) A INTEGER auxiliary array.
IW(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-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.
XDEL U DOUBLE PRECISION trust region stepsize; the choice
XDEL=0 causes that a suitable default value is
computed.
MIT U INTEGER variable that specifies the maximum number of
iterations; the choice MIT=0 causes that the default
value 5000 will be taken.
MFV U INTEGER variable that specifies the maximum number of
function evaluations; the choice MFV=0 causes that
the default value 5000 will be taken.
MFG U INTEGER variable that specifies the maximum number of
gradient evaluations; the choice MFG=0 causes that
the default value 10000 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.
MOS1 U INTEGER number of Lanczos steps for determination of
the Levenberg-Marquardt parameter; The choice MOS1=0
causes that the default value 5 will be taken.
MOS2 U INTEGER choice of preconditioning strategy:
MOS2=1 - preconditioning is not used,
MOS2=2 - preconditioning by the incomplete
Gill-Murray decomposition,
MOS2=3 - preconditioning by the incomplete
Gill-Murray decomposition combined with
preliminary solution of the preconditioned
system.
The choice MOS2=0 causes that the default value 2 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 PNEC requires the user supplied subroutines OBJ
and DOBJ which are described in Section 2.
4. Verification of the subroutines:
-----------------------------------
Subroutine PNECU can be verified and tested using the program
TNECU. This program calls the subroutines TIUS14 (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 TNECU on a PC computer with Microsoft
Power Station Fortran compiler have the following form.
NIT= 1447 NFV= 1450 NFG= 5792 F= 0.173249493E-16 G= 0.138E-06 ITERM= 3
NIT= 79 NFV= 89 NFG= 400 F= 0.169144088E-20 G= 0.382E-09 ITERM= 3
NIT= 18 NFV= 19 NFG= 114 F= 0.180692317E-09 G= 0.316E-06 ITERM= 4
NIT= 24 NFV= 25 NFG= 100 F= 269.499543 G= 0.136E-08 ITERM= 4
NIT= 11 NFV= 12 NFG= 72 F= 0.990922474E-10 G= 0.511E-06 ITERM= 4
NIT= 17 NFV= 21 NFG= 252 F= 0.166904871E-10 G= 0.898E-06 ITERM= 4
NIT= 11 NFV= 12 NFG= 72 F= 336.937181 G= 0.629E-06 ITERM= 4
NIT= 6 NFV= 11 NFG= 126 F= 761774.954 G= 0.237E-05 ITERM= 2
NIT= 7 NFV= 8 NFG= 16 F= 316.436141 G= 0.362E-08 ITERM= 4
NIT= 70 NFV= 74 NFG= 639 F= -133.630000 G= 0.221E-07 ITERM= 4
NIT= 71 NFV= 72 NFG= 432 F= 10.7765879 G= 0.237E-10 ITERM= 4
NIT= 133 NFV= 134 NFG= 536 F= 982.273617 G= 0.203E-07 ITERM= 4
NIT= 7 NFV= 8 NFG= 32 F= 0.402530175E-26 G= 0.153E-13 ITERM= 3
NIT= 2 NFV= 3 NFG= 18 F= 0.129028794E-08 G= 0.820E-06 ITERM= 4
NIT= 10 NFV= 11 NFG= 44 F= 1.92401599 G= 0.217E-06 ITERM= 4
NIT= 12 NFV= 15 NFG= 78 F= -427.404476 G= 0.894E-09 ITERM= 4
NIT= 8 NFV= 9 NFG= 54 F=-0.379921091E-01 G= 0.391E-09 ITERM= 4
NIT= 8 NFV= 9 NFG= 54 F=-0.245741193E-01 G= 0.705E-10 ITERM= 4
NIT= 7 NFV= 8 NFG= 48 F= 59.5986241 G= 0.106E-08 ITERM= 4
NIT= 10 NFV= 11 NFG= 66 F= -1.00013520 G= 0.277E-11 ITERM= 4
NIT= 11 NFV= 12 NFG= 72 F= 2.13866377 G= 0.154E-06 ITERM= 4
NIT= 46 NFV= 51 NFG= 282 F= 1.00000000 G= 0.376E-08 ITERM= 4
NITER = 2015 NFVAL = 2064 NITCG = 1182 NSUCC = 22
TIME= 0:00:02.92
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 PNECS can be verified and tested using the program
TNECS. This program calls the subroutines TIUS14 (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 TNECS on a PC computer with Microsoft Power
Station Fortran compiler have the following form.
NIT= 1436 NFV= 1439 NFG= 5748 F= 3.98662385 G= 0.138E-08 ITERM= 4
NIT= 79 NFV= 89 NFG= 400 F= 0.169144088E-20 G= 0.382E-09 ITERM= 3
NIT= 18 NFV= 19 NFG= 114 F= 0.180692317E-09 G= 0.316E-06 ITERM= 4
NIT= 24 NFV= 25 NFG= 100 F= 269.499543 G= 0.136E-08 ITERM= 4
NIT= 11 NFV= 12 NFG= 72 F= 0.990922474E-10 G= 0.511E-06 ITERM= 4
NIT= 17 NFV= 21 NFG= 252 F= 0.166904871E-10 G= 0.898E-06 ITERM= 4
NIT= 11 NFV= 12 NFG= 72 F= 336.937181 G= 0.629E-06 ITERM= 4
NIT= 6 NFV= 11 NFG= 126 F= 761774.954 G= 0.237E-05 ITERM= 2
NIT= 7 NFV= 8 NFG= 16 F= 316.436141 G= 0.362E-08 ITERM= 4
NIT= 70 NFV= 74 NFG= 639 F= -133.630000 G= 0.221E-07 ITERM= 4
NIT= 27 NFV= 31 NFG= 168 F= 86.8673060 G= 0.416E-06 ITERM= 4
NIT= 133 NFV= 134 NFG= 536 F= 982.273617 G= 0.203E-07 ITERM= 4
NIT= 7 NFV= 8 NFG= 32 F= 0.402530175E-26 G= 0.153E-13 ITERM= 3
NIT= 2 NFV= 3 NFG= 18 F= 0.129028794E-08 G= 0.820E-06 ITERM= 4
NIT= 10 NFV= 11 NFG= 44 F= 1.92401599 G= 0.217E-06 ITERM= 4
NIT= 12 NFV= 15 NFG= 78 F= -427.404476 G= 0.894E-09 ITERM= 4
NIT= 8 NFV= 9 NFG= 54 F=-0.379921091E-01 G= 0.391E-09 ITERM= 4
NIT= 8 NFV= 9 NFG= 54 F=-0.245741193E-01 G= 0.705E-10 ITERM= 4
NIT= 7 NFV= 8 NFG= 48 F= 59.5986241 G= 0.106E-08 ITERM= 4
NIT= 10 NFV= 11 NFG= 66 F= -1.00013520 G= 0.277E-11 ITERM= 4
NIT= 11 NFV= 12 NFG= 72 F= 2.13866377 G= 0.154E-06 ITERM= 4
NIT= 46 NFV= 51 NFG= 282 F= 1.00000000 G= 0.376E-08 ITERM= 4
NITER = 1960 NFVAL = 2012 NITCG = 1127 NSUCC = 22
TIME= 0:00:02.88
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.