Serdica Mathematical Journal
Volume 22, Number 3, 1996
All the papers included in this issue of Serdica Mathematical Journal are
related to the topics of well-posedness and stability of variational
problems. It was proposed to have such an issue by the organizers
of the Workshop
``Well-posed Problems and Stability in Optimization'' held at the
Centre International de Rencontres Math\'ematiques (CIRM) in Marseille
(France) during September 11-15, 1995.
When doing the printout copy of the 3-th issue of Serdica Mathematical
Journal for 1996 our technical staff has done an error putting number
267 for the first page of this issue instead of number 165. We apologize
for this mistake.
C O N T E N T S
A B S T R A C T S
SOMME PONCTUELLE D'OPERATEURS MAXIMAUX MONOTONES
Attouch, A., H. Riahi and M. Thera.
1991 Mathematics Subject Classification:
47H05, 54A20, 54C60; 26B25, 46B10.
Key words:
operateur
maximal monotone, convergence au sens des graphes,
convergence au sens de Mosco,
condition de Brezis-Crandall and Pazy.
The primary goal of this paper is to shed some light on the maximality of the
pointwise sum of two maximal monotone operators.
The interesting purpose is to
extend some recent results of Attouch,
Moudafi and Riahi on the graph-convergence
of maximal monotone operators to the more
general setting of reflexive Banach spaces. In
addition, we present some conditions which imply
the uniform Brezis-Crandall-Pazy condition.
Afterwards, we present, as a consequence, some
recent conditions which ensure the Mosco-epiconvergence
of the sum of convex proper lower
semicontinuous functions.
EPICONVERGENCE D'UNE SUITE DE SOMMES EN NIVEAUX DE
FONCTIONS CONVEXES
Traore S., M. Volle.
1991 Mathematics Subject Classification:
52A41, 90C25, 90C31.
Key words:
variational convergence, convex duality,
sensitivity analysis
We consider the problem of minimizing the max of two convex functions from
both approximation and sensitivity point of view.This lead up to study the
epiconvergence of a sequence of level sums of convex functions and the
related dual problems.
STABILITY OF SUPPORTING AND EXPOSING ELEMENTS OF
CONVEX SETS IN BANACH SPACES
Aze D., R. Lucchetti.
1991 Mathematics Subject Classification:
54A20, 52A40, 52A05; 46B20.
Key words:
convex sets,
convex functions, supported and exposed points, slice topology,
Attouch-Wets topology, convex optimization
To a convex set in a Banach space we associate a convex function
(the separating function), whose subdifferential provides useful
information on the nature of the supporting and exposed points of the
convex set. These points are shown to be also connected to the solutions of
a minimization problem involving the separating function. We investigate
some relevant properties of this function and of its conjugate in the sense
of Legendre-Fenchel. Then we highlight the connections between set
convergence, with respect to the slice and Attouch-Wets topologies, and
convergence, in the same sense, of the associated functions. Finally, by
using known results on the behaviour of the subdifferential of a convex
function under the former epigraphical perturbations, we are able to derive
stability results for the set of supported points and of supporting
and exposing functionals of a closed convex subset of a Banach space.
STABILITY OF THE ITERATION METHOD FOR NON EXPANSIVE MAPPINGS
Lemaire B.
1991 Mathematics Subject Classification:
65K10, 49M07, 90C25
Key words:
convex minimization, convergence,
iteration method, gradient method, monotone inclusions, prox method, stability.
The general iteration method for nonexpansive mappings on a Banach space
is considered. Under some assumption of fast enough convergence on the
sequence of ("almost" nonexpansive) perturbed iteration mappings, if
the basic method is $\tau$-convergent for a suitable
topology $\tau$ weaker than the norm topology,
then the perturbed method is also $\tau$-convergent.
Application is presented to the gradient-prox method for
monotone inclusions in Hilbert spaces.
SPECULATING ABOUT MOUNTAINS
Ribarska N. K., Ts. Y. Tsachev, M. I. Krastanov.
1991 Mathematics Subject Classification:
58E05.
Key words:
upper semicontinuous multivalued
mappings with compact images, deformation lemma,
mountain pass theorem.
The definition of the weak slope of
continuous functions introduced by Degiovanni
and Marzocchi and its interrelation with the notion
"steepness" of locally Lipschitz
functions are discussed. A deformation lemma and a mountain pass
theorem for usco mappings are
proved. The relation between these results and the respective ones
for lower semicontinuous functions is considered.
SUBDIFFERENTIALS OF PERFORMANCE FUNCTIONS AND CALCULUS OF
CODERIVATIVES OF SET-VALUED MAPPINGS
Ioffe A. D., Jean-Paul Penot.
1991 Mathematics Subject Classification:
49J52; 49J50, 58C20.
Key words:
set-valued mapping,
lower semicontinuous function, subdifferential, normal cone,
coderivative, marginal function.
The paper contains calculus rules for coderivatives of compositions,
sums and intersections of set-valued mappings. The types of
coderivatives considered correspond to Dini-Hadamard and limiting
Dini-Hadamard subdifferentials in Gateaux differentiable spaces,
Frechet and limiting Frechet subdifferentials in Asplund spaces
and approximate subdifferentials in arbitrary Banach spaces. The key
element of the unified approach to obtaining various calculus rules
for various types of derivatives presented in the paper are simple
formulas for subdifferentials of marginal, or performance functions.
UNIFORM CONVERGENCE OF THE NEWTON METHOD
FOR AUBIN CONTINUOUS MAPS
Dontchev A. L.
1991 Mathematics Subject Classification:
90C30, 47H04, 49M37, 65K10.
Key words:
generalized equation,
Newton's method, sequential quadratic programming.
In this paper we prove that the
Newton method applied to the generalized equation
y $\in$ f(x) + F(x)
with a C^{1} function f and a set-valued map F
acting in Banach spaces, is locally convergent uniformly in
the parameter y if and only if the map
(f+F)^{-1} is Aubin continuous at
the reference point.
We also show that the Aubin continuity actually implies uniform Q-quadratic
convergence provided that the derivative of f is Lipschitz continuous.
As an application, we give a characterization of the uniform local
Q-quadratic convergence of the sequential quadratic programming method
applied to a perturbed nonlinear program.
ON A VARIATIONAL APPROACH TO SOME QUASILINEAR PROBLEMS
Canino A.
1991 Mathematics Subject Classification:
35J65
Key words:
quasilinear elliptic problems, nonsmooth critical point theory.
We prove some multiplicity results concerning quasilinear
elliptic equations with natural growth conditions.
Techniques of nonsmooth critical point theory are employed.
PERTURBATIONS OF CRITICAL VALUES IN NONSMOOTH CRITICAL POINT
THEORY
Degiovanni M., S. Lancelotti.
1991 Mathematics Subject Classification:
58E05, 35J85.
Key words:
nonsmooth critical point theory, perturbation problems,
variational convergence, elliptic variational inequatilies.
The perturbation of critical values for continuous
functionals is studied.
An application to eigenvalue problems for variational
inequalities is provided.
NONTRIVIAL SOLUTIONS OF QUASILINEAR EQUATIONS IN BV
Marzocchi M.
1991 Mathematics Subject Classification:
58E05, 35J65.
Key words:
nonsmooth critical point theory, quasilinear equations,
area functional.
The existence of a nontrivial critical point is proved for a
functional containing an area-type term. Techniques of nonsmooth
critical point theory are applied.
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