***********************************************************************
* *
* PVAR - A PROXIMAL BUNDLE ALGORITHM FOR NONSMOOTH *
* OPTIMIZATION. *
* *
***********************************************************************
1. Introduction:
----------------
The double-precision FORTRAN 77 basic subroutine PVAR is designed
to find a close approximation to a local minimum of a nonlinear
nonsmooth function F(X) with simple bounds on variables and general
linear constraints. Here X is a vector of N variables and F(X), is
assumed to be a locally Lipschitz continuous function. 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 <= N. General linear constraints are assumed in the form
C(I) unbounded if IC(I) = 0,
CL(I) <= C(I) if IC(I) = 1,
C(I) <= CU(I) if IC(I) = 2,
CL(I) <= C(I) <= CU(I) if IC(I) = 3,
CL(I) = C(I) = CU(I) if IC(I) = 5,
where C(I) = A_I * X, 1 <= I <= NC, are linear functions. To simplify
user's work, three additional easy to use subroutines are added. They
call the basic general subroutine PVAR:
PVARU - unconstrained nonsmooth optimization,
PVARS - nonsmooth optimization with simple bounds,
PVARL - nonsmooth optimization with simple bounds and general
linear constraints.
All subroutines contain a description of formal parameters and
extensive comments. Furthermore, two test programs TVARU and TVARL are
included, which contain several test problems (see e.g. [4]). 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 methods is
given in [1], [2], [3]. 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/ NDECF,NRES,NRED,NREM,NADD,NIT,NFV,NFG,NFH
The arguments have the following meaning:
Argument Type Significance
----------------------------------------------------------------------
NDECF O Positive INTEGER variable that indicates the number of
matrix decompositions.
NRES O Positive INTEGER variable that indicates the number of
restarts.
NRED O Positive INTEGER variable that indicates the number of
reductions.
NREM O Positive INTEGER variable that indicates the number of
constraint deletions during the QP solutions.
NADD O Positive INTEGER variable that indicates the number of
constraint additions during the QP solutions.
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
subgradient evaluations.
NFH O Positive INTEGER variable that specifies the number of
Hessian evaluations.
2. Subroutines PVARU, PVARS, PVARL:
-----------------------------------
The calling sequences are
CALL PVARU(NF,NA,X,IPAR,RPAR,F,GMAX,IPRNT,ITERM)
CALL PVARS(NF,NA,NB,X,IX,XL,XU,IPAR,RPAR,F,GMAX,IPRNT,ITERM)
CALL PVARL(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,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.
NA I Nonnegative INTEGER variable that specifies the maximum
bundle dimension. The choice NA=0 causes that the
default value NA=NF+3 will be taken.
NB I Nonnegative INTEGER variable that specifies whether the
simple bounds are suppressed (NB=0) or accepted (NB>0).
NC I Nonnegative INTEGER variable that specifies the number
of linear constraints; if NC=0 the linear constraints
are suppressed.
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).
CF(NC) A DOUBLE PRECISION vector which contains values of
constraint functions (only if NC>0).
IC(NC) I On input (significant only if NC>0) INTEGER vector which
contains constraint types:
IC(K)=0 - the constraint CF(K) is not used,
IC(K)=1 - the lower constraint CF(K) >= CL(K),
IC(K)=2 - the upper constraint CF(K) <= CU(K),
IC(K)=3 - the two side constraint
CL(K) <= CF(K) <= CU(K),
IC(K)=5 - the equality constraint CF(K) = CL(K).
CL(NC) I DOUBLE PRECISION vector with lower bounds for constraint
functions (significant only if NC>0).
CU(NC) I DOUBLE PRECISION vector with upper bounds for constraint
functions (significant only if NC>0).
CG(NF*NC) I DOUBLE PRECISION matrix whose columns are normals of the
linear constraints (significant only if NC>0).
IPAR(5) A INTEGER parameters:
IPAR(1)=MIT, IPAR(2)=MFV, IPAR(3)=MEC,
IPAR(4)=MTESX, IPAR(5)=MTESF.
Parameters MIT, MFV, MEC, MTESX, MTESF are described
in Section 3 together with other parameters of the
subroutine PBUN.
RPAR(6) A DOUBLE PRECISION parameters:
RPAR(1)=XMAX, RPAR(2)=TOLX, RPAR(3)=TOLF,
RPAR(4)=TOLB, RPAR(5)=TOLG, RPAR(6)=ETA.
Parameters XMAX, TOLX, TOLF, TOLB, TOLG, ETA are
described in Section 3 together with other parameters
of the subroutine PBUN.
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 MTESX subsequent iterations,
ITERM= 2 - if |F - FO| was less than or equal to TOLF
in MTESF 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=11 - if NFV exceeded MFV,
ITERM=12 - if NIT exceeded MIT,
ITERM< 0 - if the method failed. If ITERM=-6, then the
termination criterion has not been satisfied,
but the point obtained is usually acceptable.
The subroutines PVARU, PVARS, PVARL require the user supplied subroutine
FUNDER that defines the objective function and its subgradient and has
the form
SUBROUTINE FUNDER(NF,X,F,G)
The arguments of the user supplied subroutine 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 subgradient of the objective function
at the point X.
3. Subroutine PVAR:
-------------------
This general subroutine is called from all the subroutines described
in Section 2. The calling sequence is
CALL PVAR(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,ICA,CFD,CR,CZ,
& AF,AX,AG,G,GN,H,S,SN,XO,GO,GP,GS,XMAX,TOLX,TOLF,TOLB,TOLG,ETA,
& XMAX,GMAX,F,MIT,MFV,MEX,MTESX,MTESF,IPRNT,ITERM).
The arguments NF, NA, NB, NC, X, IX, XL, XU, CF, IC, CL, CU, CG,
GMAX, F, ITERM, IPRNT have the same meaning as in Section 2. Other
arguments have the following meaning:
Argument Type Significance
----------------------------------------------------------------------
ICA(NC) A INTEGER vector containing indices of active constraints.
CFD(NC) A DOUBLE PRECISION vector of constraint function
increments.
CR(NCR) A DOUBLE PRECISION matrix containing triangular
decomposition of the orthogonal projection kernel
(NCR is equal to NF*(NF+1)/2).
CZ(NF*NF) A DOUBLE PRECISION matrix containing orthogonal basis
of linear manifold defined by active constraints.
AF(4*NA) A DOUBLE PRECISION vector of bundle function values.
AX(NF*NA) A DOUBLE PRECISION matrix whose columns are bundle
points.
AG(NF*NA) A DOUBLE PRECISION matrix whose columns are bundle
subgradients.
G(NF) A DOUBLE PRECISION subgradient of the objective function.
GN(NF) A DOUBLE PRECISION Reduced aggregated subgradient of the
objective function.
H(NH) A DOUBLE PRECISION variable metric approximation of the
Hessian matrix.
S(NF) A DOUBLE PRECISION direction vector.
SN(NF) A DOUBLE PRECISION reduced direction vector.
XO(NF) A DOUBLE PRECISION vector which contains increments of
variables.
GO(NF) A DOUBLE PRECISION vector which contains increments of
subgradients.
GP(NF) A DOUBLE PRECISION Aggregated subgradient of the objective
function.
GS(NF) A DOUBLE PRECISION Auxiliary vector.
XMAX I DOUBLE PRECISION maximum stepsize; the choice XMAX=0
causes that the default value 1.0D3 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-8 will be taken.
TOLB I DOUBLE PRECISION minimum acceptable function value;
the choice TOLB=0 causes that the default value
TOLB=-1.0D60 will be taken.
TOLG I DOUBLE PRECISION tolerance for the termination
criterion; the choice TOLG=0 causes that the default
value TOLG=1.0D-6 will be taken.
ETA I DOUBLE PRECISION distance measure parameter.
MIT I INTEGER variable that specifies the maximum number of
iterations; the choice MIT=0 causes that the default
value 200 will be taken.
MFV I INTEGER variable that specifies the maximum number of
function evaluations; the choice |MFV|=0 causes that
the default value 500 will be taken.
MEX I INTEGER version of nonsmooth variable metric method:
MEX=0 - convex version,
MET=1 - nonconvex version.
MTESX I INTEGER variable that specifies the maximum number of
iterations with changes of the coordinate vector X
smaller than TOLX; the choice MTESX=0 causes that the
default value MTESX=20 will be taken.
MTESF I INTEGER variable that specifies the maximum number of
iterations with changes of function values smaller
than TOLF; the choice MTESF=0 causes that the default
value MTESF=2 will be taken.
The subroutine PVAR has a modular structure. The following list
contains its most important subroutines:
PLLPB1 - Primal null space method for solving a linear
programming subproblem with linear constraints.
PS1L07 - Simplified line search using function values and
derivatives.
PS1L08 - Nonsmooth line search using function values and
derivatives.
PUDVI2 - Special nonsmooth variable metric updates.
The subroutine PVAR requires the user supplied subroutine FUNDER
which is described in Section 2.
4. Subroutine PLLPB1:
---------------------
Since the primal null space method for linear programming subproblems
can be used separately in many applications, we describe the subroutine
PLLPB1 in more details. The calling sequence is
CALL PLLPB1(NF,NC,X,IX,XO,XL,XU,CF,CFD,IC,ICA,CL,CU,CG,CR,
& CZ,G,GO,S,MFP,KBF,KBC,ETA9,EPS7,EPS9,UMAX,GMAX,N,ITERL)
The arguments NF, NC, X, IX, XL, XU, CF, IC, CL, CU, CG have
the same meaning as in Section 2 (only with the difference that the
arguments X and CF are of the type (I), i.e. they must have a value
defined on entry to PLLPB1 and they are not changed). The arguments CFD,
ICA, CR, CZ have the same meaning as in Section 3 (only with the
difference that the arguments CFD, ICA, CR, CZ are of the type (O),
i.e. their values can be used subsequently). Other arguments have the
following meaning:
Argument Type Significance
---------------------------------------------------------------------
G(NF) I DOUBLE PRECISION subgradient of the objective function.
XO(NF) A DOUBLE PRECISION auxiliary vector.
approximate Hessian (NH is equal to NF*(NF+1)/2).
S(NF) O DOUBLE PRECISION direction vector.
MFP I INTEGER variable that specifies the type of the
computed point.
MFP=1 - computation is terminated whenever an
arbitrary feasible point is found,
MFP=2 - computation is terminated whenever an
optimum feasible point is found,
MFP=3 - computation starts from the previously
reached point and is terminated whenever
an optimum feasible point is found.
KBF I INTEGER variable that specifies simple bounds on
variables.
KBF=0 - simple bounds are suppressed,
KBF=1 - one sided simple bounds,
KBF=2 - two sided simple bounds.
KBC I INTEGER variable that specifies general linear
constraints.
KBC=0 - linear constraints are suppressed,
KBC=1 - one sided linear constraints,
KBC=2 - two sided linear constraints.
ETA9 I DOUBLE PRECISION maximum floating point number.
EPS7 I DOUBLE PRECISION tolerance for linear independence
of constraints (the recommended value is 1.0D-10).
EPS9 I DOUBLE PRECISION tolerance for the definition of active
constraints (the recommended value is 1.0D-8).
UMAX O DOUBLE PRECISION maximum absolute value of the negative
Lagrange multiplier.
GMAX O DOUBLE PRECISION infinity norm of the gradient of the
Lagrangian function.
N O INTEGER dimension of the manifold defined by active
constraints.
ITERL O INTEGER variable that indicates the type of the
computed feasible point.
ITERL= 1 - an arbitrary feasible point was found,
ITERL= 2 - the optimum feasible point was found,
ITERL=-1 - an arbitrary feasible point does not
exist,
ITERL=-2 - the optimum feasible point does not
exist.
5. Form of printed results:
---------------------------
The form of printed results is specified by the parameter IPRNT
as is described in Section 3. Here we demonstrate individual forms of
printed results by the simple use of the program TVARU described in the
next section (with NEXT=11). If we set IPRNT=1, then the printed results
will have the form
NIT= 14 NFV= 14 NFG= 14 F= -.79999998D+01 G= .4535D-06 ITERM= 4
If we set IPRNT=-1, then the printed results will have the form
EXIT FROM PVAR :
NIT= 14 NFV= 14 NFG= 14 F= -.79999998D+01 G= .4535D-06 ITERM= 4
X= -.1000072D+01 -.1728811D-09
If we set IPRNT=2, then the printed results will have the form
ENTRY TO PVAR :
NIT= 0 NFV= 1 NFG= 1 F= .60207973D+02 G= .1329D+02
NIT= 1 NFV= 2 NFG= 2 F= .43113601D+02 G= .1229D+02
NIT= 2 NFV= 3 NFG= 3 F= .26633740D+02 G= .1346D+02
NIT= 3 NFV= 4 NFG= 4 F= .11345282D+02 G= .1495D+02
NIT= 4 NFV= 5 NFG= 5 F= .37878798D+01 G= .1600D+02
NIT= 5 NFV= 6 NFG= 6 F= -.52179782D+01 G= .1600D+02
NIT= 6 NFV= 7 NFG= 7 F= -.52179782D+01 G= .6225D+01
NIT= 7 NFV= 8 NFG= 8 F= -.65520912D+01 G= .1600D+02
NIT= 8 NFV= 9 NFG= 9 F= -.77403014D+01 G= .1600D+02
NIT= 9 NFV= 10 NFG= 10 F= -.79743187D+01 G= .1600D+02
NIT= 10 NFV= 11 NFG= 11 F= -.79987188D+01 G= .1600D+02
NIT= 11 NFV= 12 NFG= 12 F= -.79999678D+01 G= .1600D+02
NIT= 12 NFV= 13 NFG= 13 F= -.79999994D+01 G= .1600D+02
NIT= 13 NFV= 14 NFG= 14 F= -.79999998D+01 G= .1600D+02
EXIT FROM PVAR :
NIT= 14 NFV= 14 NFG= 14 F= -.79999998D+01 G= .4535D-06 ITERM= 4
If we set IPRNT=-2, then the printed results will have the form
ENTRY TO PVAR :
NIT= 0 NFV= 1 NFG= 1 F= .60207973D+02 G= .1329D+02
NIT= 1 NFV= 2 NFG= 2 F= .43113601D+02 G= .1229D+02
NIT= 2 NFV= 3 NFG= 3 F= .26633740D+02 G= .1346D+02
NIT= 3 NFV= 4 NFG= 4 F= .11345282D+02 G= .1495D+02
NIT= 4 NFV= 5 NFG= 5 F= .37878798D+01 G= .1600D+02
NIT= 5 NFV= 6 NFG= 6 F= -.52179782D+01 G= .1600D+02
NIT= 6 NFV= 7 NFG= 7 F= -.52179782D+01 G= .6225D+01
NIT= 7 NFV= 8 NFG= 8 F= -.65520912D+01 G= .1600D+02
NIT= 8 NFV= 9 NFG= 9 F= -.77403014D+01 G= .1600D+02
NIT= 9 NFV= 10 NFG= 10 F= -.79743187D+01 G= .1600D+02
NIT= 10 NFV= 11 NFG= 11 F= -.79987188D+01 G= .1600D+02
NIT= 11 NFV= 12 NFG= 12 F= -.79999678D+01 G= .1600D+02
NIT= 12 NFV= 13 NFG= 13 F= -.79999994D+01 G= .1600D+02
NIT= 13 NFV= 14 NFG= 14 F= -.79999998D+01 G= .1600D+02
EXIT FROM PVAR :
NIT= 14 NFV= 14 NFG= 14 F= -.79999998D+01 G= .4535D-06 ITERM= 4
X= -.1000072D+01 -.1728811D-09
6. Verification of the subroutines:
-----------------------------------
Subroutine PVARU can be verified and tested using the program
TVARU. This program calls the subroutines TIUD19 (initiation), TFFU19
(function evaluation) and TFGU19 (subgradient evaluation) containing
20 academic test problems with at most 50 variables [4]. The results
obtained by the program TVARU (with MEX=0) on a PC computer with
Microsoft Power Station Fortran compiler have the following form.
NIT= 56 NFV= 57 NFG= 57 F= 0.103091036E-08 G= 0.207E-08 ITERM= 4
NIT= 15 NFV= 16 NFG= 16 F= 0.948941202E-10 G= 0.474E-10 ITERM= 4
NIT= 26 NFV= 26 NFG= 26 F= 1.95222473 G= 0.779E-06 ITERM= 4
NIT= 17 NFV= 17 NFG= 17 F= 2.00000000 G= 0.292E-07 ITERM= 4
NIT= 22 NFV= 22 NFG= 22 F= -2.99999998 G= 0.402E-07 ITERM= 4
NIT= 22 NFV= 22 NFG= 22 F= 7.20000067 G= 0.492E-06 ITERM= 4
NIT= 14 NFV= 14 NFG= 14 F= -1.41421356 G= 0.175E-06 ITERM= 4
NIT= 58 NFV= 63 NFG= 63 F=-0.999999928 G= 0.103E-06 ITERM= 4
NIT= 62 NFV= 62 NFG= 62 F=-0.999999852 G= 0.574E-07 ITERM= 4
NIT= 39 NFV= 39 NFG= 39 F= -7.99999983 G= 0.848E-06 ITERM= 4
NIT= 74 NFV= 75 NFG= 75 F= -43.9999976 G= 0.433E-06 ITERM= 4
NIT= 49 NFV= 49 NFG= 49 F= 22.6001627 G= 0.241E-06 ITERM= 4
NIT= 52 NFV= 53 NFG= 53 F= -32.3486784 G= 0.100E-05 ITERM= 2
NIT= 32 NFV= 32 NFG= 32 F= -2.91970037 G= 0.341E-06 ITERM= 4
NIT= 114 NFV= 114 NFG= 114 F= 0.559818540 G= 0.624E-06 ITERM= 4
NIT= 112 NFV= 112 NFG= 112 F=-0.841396818 G= 0.474E-06 ITERM= 4
NIT= 158 NFV= 158 NFG= 158 F= 9.78600636 G= 0.890E-06 ITERM= 4
NIT= 105 NFV= 105 NFG= 105 F= 16.7038379 G= 0.446E-06 ITERM= 4
NIT= 128 NFV= 129 NFG= 129 F= 0.160715615E-05 G= 0.355E-06 ITERM= 4
NIT= 22 NFV= 22 NFG= 22 F= 0.00000000 G= 0.000E+00 ITERM= 4
NITER = 1177 NFVAL = 1187 NSUCC = 20
TIME= 0:00:00.03
The rows corresponding to individual test problems contain the number of
iterations NIT, the number of function evaluations NFV, the number of
subgradient evaluations NFG, the final value of the objective function F,
the value of the criterion for the termination G and the cause of
termination ITERM.
Subroutine PVARL can be verified and tested using the program
TVARL. This program calls the subroutines TIUD22 (initiation), TAFU22
(function evaluation), TAGU22 (subgradient evaluation) containing
15 academic test problems with at most 20 variables [4]. The results
obtained by the program TVARL on a PC computer with Microsoft Power
Station Fortran compiler have the following form.
NIT= 11 NFV= 11 NFG= 11 F=-0.389659516 G= 0.163E-07 ITERM= 4
NIT= 5 NFV= 8 NFG= 8 F=-0.330357143 G= 0.487E-30 ITERM= 2
NIT= 27 NFV= 27 NFG= 27 F=-0.448910784 G= 0.416E-06 ITERM= 4
NIT= 86 NFV= 86 NFG= 86 F=-0.429280615 G= 0.297E-07 ITERM= 4
NIT= 24 NFV= 24 NFG= 24 F= -1.85961862 G= 0.294E-06 ITERM= 4
NIT= 25 NFV= 25 NFG= 25 F= 0.101830939 G= 0.943E-07 ITERM= 4
NIT= 94 NFV= 94 NFG= 94 F= 0.526114262E-05 G= 0.777E-06 ITERM= 4
NIT= 159 NFV= 159 NFG= 159 F= 24.3064637 G= 0.652E-06 ITERM= 4
NIT= 226 NFV= 227 NFG= 227 F= 133.728285 G= 0.174E-05 ITERM= 2
NIT= 144 NFV= 145 NFG= 145 F= 0.506950749 G= 0.598E-05 ITERM= 2
NIT= 658 NFV= 658 NFG= 658 F= 0.297444605E-03 G= 0.727E-06 ITERM= 4
NIT= 203 NFV= 212 NFG= 212 F= -1768.57476 G= 0.210E-04 ITERM= 2
NIT= 451 NFV= 448 NFG= 448 F= 1227.28556 G= 0.143E-04 ITERM= 2
NIT= 294 NFV= 479 NFG= 479 F= 7049.25599 G= 0.217E-03 ITERM= 2
NIT= 181 NFV= 180 NFG= 180 F= 174.787082 G= 0.374E-03 ITERM= 2
NITER = 2588 NFVAL = 2783 NSUCC = 15
TIME= 0:00:00.06
References:
-----------
[1] Luksan L., Vlcek J.: Globally convergent variable metric method
for convex nonsmooth unconstrained minimization. Journal of
Optimization Theory and Applications Vol.102, 1999, pp.593-613.
[2] Vlcek J., Luksan L.: Globally convergent variable metric method
for nonconvex nondifferentiable unconstrained minimization.
Technical Report B 8/1999. Department of Mathematical Information
Technology. University of Jyvaskyla, 1999.
[3] Luksan L., Vlcek J.: NDA: Algorithms for Nondifferentiable
Optimization. Research Report V-797, Institute of Computer Science,
Academy of Sciences of the Czech Republic, Prague, Czech Republic,
2000.
[4] Luksan L., Vlcek J.: Subroutines for Testing Nonsmooth
Unconstrained and Linearly Constrained Optimization Problems.
Research Report V-798, Institute of Computer Science,
Academy of Sciences of the Czech Republic, Prague, Czech Republic,
2000.