***********************************************************************
* *
* PGAD - HYBRID GAUSS-NEWTON METHOD WITH SECOND-ORDER CORRECTIONS *
* AND DIRECT DECOMPOSITION TRUST-REGION SUBALGORITHMS FOR *
* LARGE-SCALE PARTIALLY SEPARABLE LEAST SQUARES PROBLEMS. *
* *
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1. Introduction:
----------------
The double-precision FORTRAN 77 basic subroutine PGAD is designed
to find a close approximation to a local minimum of a sum of squares
F(X) = FA_1(X)**2 + FA_2(X)**2 + ... + FA_NA(X)**2
with simple bounds on variables. Here X is a vector of NF variables and
FA_I(X), 1 <= I <= NA, are twice continuously differentiable 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 mapping AF(X) = [FA_1(X), FA_2(X), ..., FA_NA(X)] has a sparse
Jacobian matrix, which will be denoted by AG(X) (it has NA rows and NF
columns). 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 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, two additional easy to use subroutines
are added. They call the basic general subroutine PGAD:
PGADU - unconstrained large-scale optimization,
PGADS - large-scale optimization with simple bounds.
All subroutines contain a description of formal parameters and
extensive comments. Furthermore, two test programs TGADU and TGADS 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 PGADU, PGADS:
----------------------------
The calling sequences are
CALL PGADU(NF,NA,MA,X,AF,IAG,JAG,IPAR,RPAR,F,GMAX,IDER,ISPAS,IPRNT,
& ITERM)
CALL PGADS(NF,NA,MA,X,IX,XL,XU,AF,IAG,JAG,IPAR,RPAR,F,GMAX,IDER,
& 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.
IX(NF) I On input (significant only for PGADS) 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 PGADS).
XU(NF) I DOUBLE PRECISION vector with upper bounds for variables
(significant only for PGADS).
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)=MFG,
IPAR(4)=MEC, IPAR(5)=MOS, IPAR(6)-unused,
IPAR(7)=IFIL.
Parameters MIT, MFV, MFG, MEC, MOS are described in
Section 3 together with other parameters of the
subroutine PGAD. 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)=ETA, RPAR(9)-unused.
Parameters XMAX, TOLX, TOLF, TOLB, TOLG, FMIN, XDEL,
ETA are described in Section 3 together with other
parameters of the subroutine PGAD.
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.
IDER I INGEGER variable that specifies the degree of analytically
computed derivatives (0 OR 1).
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=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 PGADU, PGADS require the user supplied subroutines
FUN and DFUN that define partial functions and their gradients and have
the form
SUBROUTINE FUN(NF,KA,X,FA)
SUBROUTINE DFUN(NF,KA,X,GA)
If IDER=0, the subroutine DFUN can be empty. 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 gradient of the KA-th partial function
at the point X. Note that only nonzero elements of this
gradient have to be assigned.
3. Subroutine PGAD:
-------------------
This general subroutine is called from all subroutines described
in Section 2. The calling sequence is
CALL PGAD(NF,NA,NB,MMAX,X,IX,XL,XU,AF,GA,AG,G,HA,AH,H,IH,JH,IAG,
& JAG,S,XO,GO,AGO,XS,PSL,PERM,INVP,WN11,WN12,WN13,WN14,XMAX,TOLX,
& TOLF,TOLB,TOLG,FMIN,XDEL,ETA,GMAX,F,MIT,MFV,MFG,MEC,MOS,IDER,
& IPRNT,ITERM)
The arguments NF, NA, X, IX, XL, XU, AF, IAG, JAG, GMAX, F, IDER, IPRNT,
ITERM have the same meaning as in Section 2. Other arguments have the
following meaning:
Argument Type Significance
----------------------------------------------------------------------
NB I Nonnegative INTEGER variable that specifies whether the
simple bounds are suppressed (NB=0) or accepted (NB>0).
MMAX I INTEGER size of array H.
GA(NF) A DOUBLE PRECISION gradient of the partial function.
AG(MA) A DOUBLE PRECISION nonzero elements of the Jacobian
matrix. This array is used only if MEC=3.
G(NF) A DOUBLE PRECISION gradient of the objective function.
HA(ML) A DOUBLE PRECISION Hessian matrix of the partial function.
AH(MH) A DOUBLE PRECISION approximation of the partitioned
Hessian matrix.
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.
AGO(MA) A DOUBLE PRECISION difference between the current and the
old Jacobian matrices. This array is used only if MEC=3.
XS(NF) 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-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; the choice FMIN<0 causes that the default value
value TOLG=0.0D 0 will be taken.
XDEL U DOUBLE PRECISION trust region stepsize; the choice
XDEL=0 causes that a suitable default value is
computed.
ETA U DOUBLE PRECISION parameter for switch between the
Gauss-Newton method and variable metric correction;
the choice ETA=0 causes that the default value
ETA=1.5D-4 will be taken.
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.
MEC U INTEGER method of a second order correction:
MEC=1 - correction by the Marwil sparse variable
metric update,
MEC=2 - correction by differences of gradients
(discrete Newton correction).
MEC=3 - correction by the Griewank-Toint partitioned
variable metric update (symmetric rank-one).
This correction uses three additional
matrices (arrays AG, AGO and AH).
The choice MEC=0 causes that the default value 2 will
be taken.
MOS U INTEGER method for computing trust-region step:
MOS=1 - double dog-leg method of Dennis and Mei,
MOS=2 - method of More and Sorensen for obtaining
optimum locally constrained step.
The choice MOS=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 PGAD requires the user supplied subroutines FUN
and DFUN which are described in Section 2.
4. Verification of the subroutines:
-----------------------------------
Subroutine PGADU can be verified and tested using the program
TGADU. This program calls the subroutines TIUB15 (initiation), TAFU15
(function evaluation) and TAGU15 (gradient evaluation) containing
22 unconstrained test problems with at most 1000 variables [2]. The
results obtained by the program TGADU on a PC computer with Microsoft
Power Station Fortran compiler have the following form.
NIT= 1377 NFV= 1379 NFG= 1379 F= 0.697391982E-22 G= 0.130E-09 ITERM= 3
NIT= 41 NFV= 46 NFG= 46 F= 0.216572157E-16 G= 0.154E-06 ITERM= 3
NIT= 11 NFV= 12 NFG= 14 F= 0.136731713E-09 G= 0.233E-06 ITERM= 4
NIT= 13 NFV= 16 NFG= 21 F= 134.749772 G= 0.279E-06 ITERM= 4
NIT= 4 NFV= 5 NFG= 7 F= 0.111058357E-10 G= 0.887E-06 ITERM= 4
NIT= 6 NFV= 7 NFG= 13 F= 0.742148235E-26 G= 0.303E-12 ITERM= 3
NIT= 10 NFV= 12 NFG= 23 F= 60734.8551 G= 0.648E-07 ITERM= 4
NIT= 21 NFV= 26 NFG= 24 F= 0.253357740E-08 G= 0.800E-06 ITERM= 4
NIT= 15 NFV= 16 NFG= 36 F= 2216.45871 G= 0.104E-10 ITERM= 4
NIT= 12 NFV= 18 NFG= 21 F= 191.511336 G= 0.524E-07 ITERM= 4
NIT= 2587 NFV= 2593 NFG= 2649 F= 0.647358980E-27 G= 0.359E-12 ITERM= 3
NIT= 16 NFV= 20 NFG= 23 F= 19264.6341 G= 0.513E-10 ITERM= 4
NIT= 17 NFV= 21 NFG= 28 F= 131234.018 G= 0.784E-08 ITERM= 4
NIT= 5 NFV= 8 NFG= 18 F= 108.517888 G= 0.227E-08 ITERM= 4
NIT= 6 NFV= 7 NFG= 15 F= 18.1763146 G= 0.290E-06 ITERM= 4
NIT= 15 NFV= 21 NFG= 40 F= 2.51109677 G= 0.724E-06 ITERM= 4
NIT= 15 NFV= 20 NFG= 19 F= 0.257973699E-16 G= 0.275E-08 ITERM= 3
NIT= 42 NFV= 44 NFG= 45 F= 0.151517993E-24 G= 0.122E-10 ITERM= 3
NIT= 15 NFV= 16 NFG= 23 F= 0.354943701E-14 G= 0.255E-06 ITERM= 4
NIT= 26 NFV= 27 NFG= 29 F= 0.378161520E-10 G= 0.407E-07 ITERM= 4
NIT= 10 NFV= 11 NFG= 17 F= 647.828517 G= 0.773E-11 ITERM= 4
NIT= 26 NFV= 32 NFG= 45 F= 4486.97024 G= 0.602E-07 ITERM= 4
NITER = 4290 NFVAL = 4357 NSUCC = 22
TIME= 0:00:04.56
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 PGADS can be verified and tested using the program
TGADS. This program calls the subroutines TIUB15 (initiation), TAFU15
(function evaluation), TAGU15 (gradient evaluation) containing 22 box
constrained test problems with at most 1000 variables [2]. The results
obtained by the program TGADS on a PC computer with Microsoft Power
Station Fortran compiler have the following form.
NIT= 1011 NFV= 1013 NFG= 1013 F= 0.00000000 G= 0.000E+00 ITERM= 3
NIT= 260 NFV= 273 NFG= 508 F= 1959.28649 G= 0.439E-12 ITERM= 4
NIT= 10 NFV= 12 NFG= 13 F= 0.784354965E-09 G= 0.868E-06 ITERM= 4
NIT= 14 NFV= 18 NFG= 19 F= 134.761343 G= 0.827E-08 ITERM= 4
NIT= 4 NFV= 5 NFG= 7 F= 0.438081882E-11 G= 0.697E-06 ITERM= 4
NIT= 6 NFV= 7 NFG= 13 F= 0.791460684E-17 G= 0.934E-08 ITERM= 3
NIT= 22 NFV= 23 NFG= 61 F= 145814.000 G= 0.000E+00 ITERM= 4
NIT= 25 NFV= 32 NFG= 28 F= 0.978141069E-06 G= 0.782E-06 ITERM= 4
NIT= 44 NFV= 45 NFG= 153 F= 2220.17880 G= 0.181E-09 ITERM= 4
NIT= 12 NFV= 19 NFG= 21 F= 191.511336 G= 0.301E-07 ITERM= 4
NIT= 3977 NFV= 2992 NFG= 2990 F= 0.00000000 G= 0.000E+00 ITERM= 3
NIT= 29 NFV= 30 NFG= 50 F= 67887.2385 G= 0.438E-12 ITERM= 4
NIT= 19 NFV= 20 NFG= 36 F= 147906.000 G= 0.000E+00 ITERM= 4
NIT= 1 NFV= 2 NFG= 6 F= 126.690556 G= 0.000E+00 ITERM= 4
NIT= 24 NFV= 27 NFG= 81 F= 18.1763146 G= 0.203E-10 ITERM= 4
NIT= 46 NFV= 50 NFG= 135 F= 3.59074140 G= 0.470E-10 ITERM= 4
NIT= 11 NFV= 12 NFG= 15 F= 0.969524252E-21 G= 0.171E-10 ITERM= 3
NIT= 0 NFV= 1 NFG= 3 F= 0.00000000 G= 0.000E+00 ITERM= 3
NIT= 26 NFV= 30 NFG= 34 F= 0.202602070E-14 G= 0.193E-06 ITERM= 4
NIT= 929 NFV= 930 NFG= 2780 F= 498.800124 G= 0.359E-05 ITERM= 2
NIT= 20 NFV= 21 NFG= 33 F= 649.598077 G= 0.280E-08 ITERM= 4
NIT= 24 NFV= 31 NFG= 55 F= 4488.96148 G= 0.242E-07 ITERM= 4
NITER = 6514 NFVAL = 5593 NSUCC = 22
TIME= 0:00:07.99
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.