***********************************************************************
* *
* PBUN - A PROXIMAL BUNDLE ALGORITHM FOR NONSMOOTH *
* OPTIMIZATION. *
* *
***********************************************************************
1. Introduction:
----------------
The double-precision FORTRAN 77 basic subroutine PBUN 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 PBUN:
PBUNU - unconstrained nonsmooth optimization,
PBUNS - nonsmooth optimization with simple bounds,
PBUNL - nonsmooth optimization with simple bounds and general
linear constraints.
All subroutines contain a description of formal parameters and
extensive comments. Furthermore, two test programs TBUNU and TBUNL are
included, which contain several test problems (see [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 [2] and [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 PBUNU, PBUNS, PBUNL:
-----------------------------------
The calling sequences are
CALL PBUNU(NF,NA,X,IPAR,RPAR,F,GMAX,IPRNT,ITERM)
CALL PBUNS(NF,NA,NB,X,IX,XL,XU,IPAR,RPAR,F,GMAX,IPRNT,ITERM)
CALL PBUNL(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)=MEX,
IPAR(4)=MTESX, IPAR(5)=MTESF.
Parameters MIT, MFV, MEX, 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 PBUNU, PBUNS, PBUNL 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 PBUN:
-------------------
This general subroutine is called from all the subroutines described
in Section 2. The calling sequence is
CALL PBUN(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,AF,IA,AFD,AG,
& IAA,AR,AZ,G,H,S,XO,GO,XS,GS,XMAX,TOLX,TOLF,TOLB,TOLG,ETA,
& GMAX,F,MIT,MFV,MET,MTESX,MTESF,IPRNT,ITERM).
The arguments NF, NA, NB, NC, X, IX, XL, XU, CF, IC, CL, CU, CG,
GMAX, F, IPRNT, ITERM have the same meaning as in Section 2. Other
arguments have the following meaning:
Argument Type Significance
---------------------------------------------------------------------
AF(4*NA) A DOUBLE PRECISION vector of bundle function values.
IA(NA) A INTEGER vector containing types of bundle functions.
AFD(NA) A DOUBLE PRECISION vector of bundle function increments.
AG(NF*NA) A DOUBLE PRECISION matrix whose columns are bundle
subgradients.
IAA(NA) A INTEGER vector containing indices of active functions.
AR(NAR) A DOUBLE PRECISION matrix containing triangular
decomposition of the orthogonal projection kernel
(NAR is equal to NF*(NF+1)/2).
AZ(NF+1) A DOUBLE PRECISION vector of Lagrange multipliers.
G(NF) A DOUBLE PRECISION subgradient of the objective function.
H(NF) A DOUBLE PRECISION diagonal matrix of weight parameters.
S(NF+1) A DOUBLE PRECISION direction vector.
XO(NF) A DOUBLE PRECISION vector which contains increments of
variables.
GO(NF+1) A DOUBLE PRECISION gradient of the Lagrangian function.
XS(NF) A DOUBLE PRECISION auxiliary vector.
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.
MET I INTEGER variable that specifies the weight updating
method:
MET=1 - quadratic interpolation,
MET=2 - local minimization,
MET=3 - quasi-Newton condition.
The choice MET=0 causes that the default value MET=1
will be taken.
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 PBUN has a modular structure. The following list
contains its most important subroutines:
PDDBQ1 - Determination of the descent direction using quadratic
programming subroutine and bundle updating.
PLQDF1 - Dual range space method for solving a quadratic
programming subproblem with linear constraints
(see [1]).
PS1L05 - Line search using function values and derivatives.
The subroutine PBUN requires the user supplied subroutine FUNDER
which is described in Section 2.
4. Subroutine PLQDF1:
---------------------
Since the dual range space method for special quadratic programming
subproblems arising in bundle type nonsmooth optimization can be used
separately in many applications (e.g. in minimax optimization), we
describe the subroutine PLQDF1 in more details. The calling sequence
is
CALL PLQDF1(NF,NA,NC,X,IX,XL,XU,AF,AFD,IA,IAA,AG,AR,AZ,
& CF,IC,CL,CU,CG,G,H,S,MFP,KBF,KBC,IDECF,ETA0,ETA2,ETA9,
& EPS7,EPS9,XNORM,UMAX,GMAX,N,ITERQ)
The arguments NF, NA, NC, X, IX, XL, XU, AF, CF, IC, CL, CU, CG have
the same meaning as in Section 2 (only with the difference that the
arguments X and AF are of the type (I), i.e. they must have a value
defined on entry to PLQDF1 and they are not changed). The arguments AFD,
IA, IAA, AG, AR, AZ have the same meaning as in Section 3 (only with
the difference that the arguments AFD, IAA, AR, AZ are of the type (O),
i.e. their values can be used subsequently). Other arguments have the
following meaning:
Argument Type Significance
---------------------------------------------------------------------
G(NF+1) O DOUBLE PRECISION gradient of the Lagrangian function.
H(NH) U DOUBLE PRECISION Choleski decomposition of the
approximate Hessian (NH is equal to NF*(NF+1)/2).
S(NF+1) 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.
IDECF U INTEGER variable that specifies the type of matrix
decomposition.
IDECF= 0 - no decomposition,
IDECF= 1 - Choleski decomposition,
IDECF= 9 - inversion,
IDECF=10 - diagonal matrix.
ETA0 I DOUBLE PRECISION machine precision (the recommended
value is 1.0D-15.
ETA2 I DOUBLE PRECISION tolerance for positive definiteness
in the Choleski decomposition.
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).
XNORM O DOUBLE PRECISION value of the linearized minimax
function.
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 a manifold defined by active
constraints.
ITERQ O INTEGER variable that indicates the type of the
computed feasible point.
ITERQ= 1 - an arbitrary feasible point was found,
ITERQ= 2 - the optimum feasible point was found,
ITERQ=-1 - an arbitrary feasible point does not
exist,
ITERQ=-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 TBUNU described in the
next section (with NEXT=9). If we set IPRNT=1, then the printed results
will have the form
NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 ITERM= 4
If we set IPRNT=-1, then the printed results will have the form
EXIT FROM PBUN :
NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 ITERM= 4
X= .1000000D+01 .0000000D+00
If we set IPRNT=2, then the printed results will have the form
ENTRY TO PBUN :
NIT= 0 NFV= 1 NFG= 1 F= .00000000D+00 G= .1000D+61
NIT= 1 NFV= 3 NFG= 3 F= -.37888889D+00 G= .8500D+01
NIT= 2 NFV= 4 NFG= 4 F= -.60615144D+00 G= .9333D+00
NIT= 3 NFV= 5 NFG= 5 F= -.60615144D+00 G= .8024D+00
NIT= 4 NFV= 6 NFG= 6 F= -.72848266D+00 G= .8024D+00
NIT= 5 NFV= 7 NFG= 7 F= -.72848266D+00 G= .3478D+00
NIT= 6 NFV= 8 NFG= 8 F= -.82757096D+00 G= .7222D+00
NIT= 7 NFV= 9 NFG= 9 F= -.84360358D+00 G= .1618D+00
NIT= 8 NFV= 10 NFG= 10 F= -.99860813D+00 G= .9839D-01
NIT= 9 NFV= 11 NFG= 11 F= -.99860813D+00 G= .1141D+00
NIT= 10 NFV= 12 NFG= 12 F= -.99928519D+00 G= .5014D+00
NIT= 11 NFV= 13 NFG= 13 F= -.99999999D+00 G= .5301D-01
NIT= 12 NFV= 14 NFG= 14 F= -.99999999D+00 G= .2704D-06
NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07
EXIT FROM PBUN :
NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 ITERM= 4
If we set IPRNT=-2, then the printed results will have the form
ENTRY TO PBUN :
NIT= 0 NFV= 1 NFG= 1 F= .00000000D+00 G= .1000D+61
NIT= 1 NFV= 3 NFG= 3 F= -.37888889D+00 G= .8500D+01
NIT= 2 NFV= 4 NFG= 4 F= -.60615144D+00 G= .9333D+00
NIT= 3 NFV= 5 NFG= 5 F= -.60615144D+00 G= .8024D+00
NIT= 4 NFV= 6 NFG= 6 F= -.72848266D+00 G= .8024D+00
NIT= 5 NFV= 7 NFG= 7 F= -.72848266D+00 G= .3478D+00
NIT= 6 NFV= 8 NFG= 8 F= -.82757096D+00 G= .7222D+00
NIT= 7 NFV= 9 NFG= 9 F= -.84360358D+00 G= .1618D+00
NIT= 8 NFV= 10 NFG= 10 F= -.99860813D+00 G= .9839D-01
NIT= 9 NFV= 11 NFG= 11 F= -.99860813D+00 G= .1141D+00
NIT= 10 NFV= 12 NFG= 12 F= -.99928519D+00 G= .5014D+00
NIT= 11 NFV= 13 NFG= 13 F= -.99999999D+00 G= .5301D-01
NIT= 12 NFV= 14 NFG= 14 F= -.99999999D+00 G= .2704D-06
NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07
EXIT FROM PBUN :
NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 ITERM= 4
X= .1000000D+01 .0000000D+00
6. Verification of the subroutines:
-----------------------------------
Subroutine PBUNU can be verified and tested using the program
TBUNU. 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 TBUNU on a PC computer with Microsoft Power
Station Fortran compiler have the following form.
NIT= 45 NFV= 48 NFG= 48 F= 0.198400179E-07 G= 0.325E-03 ITERM= 4
NIT= 29 NFV= 31 NFG= 31 F= 0.435801362E-13 G= 0.354E-06 ITERM= 4
NIT= 31 NFV= 33 NFG= 33 F= 1.95222452 G= 0.445E-03 ITERM= 2
NIT= 13 NFV= 15 NFG= 15 F= 2.00000000 G= 0.192E-06 ITERM= 4
NIT= 17 NFV= 19 NFG= 19 F= -3.00000000 G= 0.974E-10 ITERM= 2
NIT= 13 NFV= 15 NFG= 15 F= 7.20000149 G= 0.221E-02 ITERM= 4
NIT= 18 NFV= 21 NFG= 21 F= -1.41421356 G= 0.370E-11 ITERM= 2
NIT= 54 NFV= 56 NFG= 56 F=-0.999999830 G= 0.535E-03 ITERM= 2
NIT= 13 NFV= 15 NFG= 15 F= -1.00000000 G= 0.986E-07 ITERM= 4
NIT= 44 NFV= 47 NFG= 47 F= -7.99999992 G= 0.469E-02 ITERM= 2
NIT= 43 NFV= 45 NFG= 45 F= -43.9999991 G= 0.373E-02 ITERM= 2
NIT= 27 NFV= 29 NFG= 29 F= 22.6001621 G= 0.145E-03 ITERM= 4
NIT= 59 NFV= 61 NFG= 61 F= -32.3486789 G= 0.243E-03 ITERM= 2
NIT= 115 NFV= 116 NFG= 116 F= -2.91969280 G= 0.137E-02 ITERM= 2
NIT= 155 NFV= 156 NFG= 156 F= 0.559814839 G= 0.123E-02 ITERM= 2
NIT= 74 NFV= 75 NFG= 75 F=-0.841408290 G= 0.729E-03 ITERM= 2
NIT= 146 NFV= 148 NFG= 148 F= 9.78593906 G= 0.493E-02 ITERM= 2
NIT= 128 NFV= 151 NFG= 151 F= 16.7038653 G= 0.734E-02 ITERM= 2
NIT= 148 NFV= 149 NFG= 149 F= 0.167123812E-06 G= 0.781E-04 ITERM= 2
NIT= 39 NFV= 40 NFG= 40 F= 0.272746651E-12 G= 0.125E+00 ITERM= 2
NITER = 1211 NFVAL = 1270 NSUCC = 20
TIME= 0:00:00.06
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 PBUNL can be verified and tested using the program
TBUNL. 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 TBUNL on a PC computer with Microsoft Power
Station Fortran compiler have the following form.
NIT= 10 NFV= 11 NFG= 11 F=-0.389659516 G= 0.453E-04 ITERM= 4
NIT= 4 NFV= 5 NFG= 5 F=-0.330357143 G= 0.389E-14 ITERM= 4
NIT= 8 NFV= 10 NFG= 10 F=-0.448910786 G= 0.698E-03 ITERM= 4
NIT= 84 NFV= 85 NFG= 85 F=-0.429280614 G= 0.139E-07 ITERM= 2
NIT= 16 NFV= 17 NFG= 17 F= -1.85961382 G= 0.202E-10 ITERM= 2
NIT= 16 NFV= 17 NFG= 17 F= 0.101830889 G= 0.343E-06 ITERM= 2
NIT= 54 NFV= 57 NFG= 57 F= 0.108743445E-08 G= 0.144E-08 ITERM= 2
NIT= 73 NFV= 75 NFG= 75 F= 24.3062154 G= 0.456E-02 ITERM= 4
NIT= 148 NFV= 151 NFG= 151 F= 133.728344 G= 0.194E-01 ITERM= 2
NIT= 81 NFV= 82 NFG= 82 F= 0.506947996 G= 0.264E-06 ITERM= 2
NIT= 568 NFV= 733 NFG= 733 F= 0.604084510E-03 G= 0.687E-02 ITERM= 2
NIT= 231 NFV= 233 NFG= 233 F= -1687.55166 G= 0.456E-01 ITERM= 2
NIT= 128 NFV= 130 NFG= 130 F= 1227.22963 G= 0.210E-01 ITERM= 2
NIT= 201 NFV= 203 NFG= 203 F= 7051.22733 G= 0.287E-01 ITERM= 2
NIT= 390 NFV= 393 NFG= 393 F= 174.791184 G= 0.340E-01 ITERM= 2
NITER = 2012 NFVAL = 2202 NSUCC = 15
TIME= 0:00:00.11
References:
-----------
[1] Luksan L.: Dual Method for Solving a Special Problem of Quadratic
Programming as a Subproblem at Linearly Constrained Nonlinear
Minimax Approximation. Kybernetika 20 (1984) 445-457.
[2] Vlcek J.: Bundle Algorithms for Nonsmooth Unconstrained
Minimization. Research Report V-608, Institute of Computer Science,
Academy of Sciences of the Czech Republic, Prague, Czech Republic,
1995.
[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.