4,014 research outputs found
Solution analysis for a class of set-inclusive generalized equations: a convex analysis approach
In the present paper, classical tools of convex analysis are used to study
the solution set to a certain class of set-inclusive generalized equations. A
condition for the solution existence and global error bounds is established, in
the case the set-valued term appearing in the generalized equation is concave.
A functional characterization of the contingent cone to the solution set is
provided via directional derivatives. Specializations of these results are also
considered when outer prederivatives can be employed
On the Polyak convexity principle and its application to variational analysis
According to a result due to B.T. Polyak, a mapping between Hilbert spaces,
which is around a regular point, carries a ball centered at that
point to a convex set, provided that the radius of the ball is small enough.
The present paper considers the extension of such result to mappings defined on
a certain subclass of uniformly convex Banach spaces. This enables one to
extend to such setting a variational principle for constrained optimization
problems, already observed in finite dimension, that establishes a convex
behaviour for proper localizations of them. Further variational consequences
are explored.Comment: 13 page
Convexity of the images of small balls through perturbed convex multifunctions
In the present paper, the following convexity principle is proved: any closed
convex multifunction, which is metrically regular in a certain uniform sense
near a given point, carries small balls centered at that point to convex sets,
even if it is perturbed by adding C^{1,1} smooth mappings with controlled
Lipschizian behaviour. This result, which is valid for mappings defined on a
subclass of uniformly convex Banach spaces, can be regarded as a set-valued
generalization of the Polyak convexity principle. The latter, indeed, can be
derived as a special case of the former. Such an extension of that principle
enables one to build large classes of nonconvex multifunctions preserving the
convexity of small balls. Some applications of this phenomenon to the theory of
set-valued optimization are proposed and discussed
Generalized Benders Decomposition for one Class of MINLPs with Vector Conic Constraint
In this paper, we mainly study one class of mixed-integer nonlinear
programming problems (MINLPs) with vector conic constraint in Banach spaces.
Duality theory of convex vector optimization problems applied to this class of
MINLPs is deeply investigated. With the help of duality, we use the generalized
Benders decomposition method to establish an algorithm for solving this MINLP.
Several convergence theorems on the algorithm are also presented. The
convergence theorems generalize and extend the existing results on MINLPs in
finite dimension spaces.Comment: 20 page
Vector Equilibrium Problems on Dense Sets
In this paper we provide sufficient conditions that ensure the existence of
the solution of some vector equilibrium problems in Hausdorff topological
vector spaces ordered by a cone. The conditions that we consider are imposed
not on the whole domain of the operators involved, but rather on a self
segment-dense subset of it, a special type of dense subset. We apply the
results obtained to vector optimization and vector variational inequalities.Comment: arXiv admin note: substantial text overlap with arXiv:1405.232
On a class of convex sets with convex images and its application to nonconvex optimization
In the present paper, conditions under which the images of uniformly convex
sets through regular mappings between Banach spaces remain convex are
established. These conditions are expressed by a certain quantitative relation
betweeen the modulus of convexity of a given set and the global regularity
behaviour of the mapping on it. Such a result enables one to extend to a wide
subclass of convex sets the Polyak's convexity principle, which was originally
concerned with images of small balls around points of Hilbert spaces. In
particular, the crucial phenomenon of the preservation of convexity under
regular transformations is shown to include the class of -convex
sets, where the value of depends on the regularity behaviour of the
involved transformation. Two consequences related to nonconvex optimization are
discussed: the first one is a sufficient condition for the global solution
existence for infinite-dimensional constrained extremum problems; the second
one provides a zero-order Lagrangian type characterization of optimality in
nonlinear mathematical programming.Comment: This paper has been withdrawn by the author due to errors found in a
proo
Subdifferentials of Nonconvex Integral Functionals in Banach Spaces with Applications to Stochastic Dynamic Programming
The paper concerns the investigation of nonconvex and nondifferentiable
integral functionals on general Banach spaces, which may not be reflexive
and/or separable. Considering two major subdifferentials of variational
analysis, we derive nonsmooth versions of the Leibniz rule on
subdifferentiation under the integral sign, where the integral of the
subdifferential set-valued mappings generated by Lipschitzian integrands is
understood in the Gelfand sense. Besides examining integration over complete
measure spaces and also over those with nonatomic measures, our special
attention is drawn to a stronger version of measure nonatomicity, known as
saturation, to invoke the recent results of the Lyapunov convexity theorem type
for the Gelfand integral of the subdifferential mappings. The main results are
applied to the subdifferential study of the optimal value functions and
deriving the corresponding necessary optimality conditions in nonconvex
problems of stochastic dynamic programming with discrete time on the infinite
horizon
An extension of the Polyak convexity principle with application to nonconvex optimization
The main problem considered in the present paper is to single out classes of
convex sets, whose convexity property is preserved under nonlinear smooth
transformations. Extending an approach due to B.T. Polyak, the present study
focusses on the class of uniformly convex subsets of Banach spaces. As a main
result, a quantitative condition linking the modulus of convexity of such kind
of set, the regularity behaviour around a point of a nonlinear mapping and the
Lipschitz continuity of its derivative is established, which ensures the images
of uniformly convex sets to remain uniformly convex. Applications of the
resulting convexity principle to the existence of solutions, their
characterization and to the Lagrangian duality theory in constrained nonconvex
optimization are then discussed
Vector Representation of Preferences on -Algebras and Fair Division in Saturated Measure Spaces
The purpose of this paper is twofold. First, we axiomatize preference
relations on a -algebra of a saturated measure space represented by a
vector measure and furnish a utility representation in terms of a nonadditive
measure satisfying the appropriate requirement of continuity and convexity.
Second, we investigate the fair division problems in which each individual has
nonadditive preferences on a -algebra invoking our utility
representation result. We show the existence of individually rational Pareto
optimal partitions, Walrasian equilibria, core partitions, and Pareto optimal
envy-free partitions
Math-Selfie
This is a write up on some sections of convex geometry, functional analysis,
optimization, and nonstandard models that attract the author.Comment: 8 page
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