314 research outputs found
Quantitative Stability of Linear Infinite Inequality Systems under Block Perturbations with Applications to Convex Systems
The original motivation for this paper was to provide an efficient
quantitative analysis of convex infinite (or semi-infinite) inequality systems
whose decision variables run over general infinite-dimensional (resp.
finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed
set . Parameter perturbations on the right-hand side of the inequalities are
required to be merely bounded, and thus the natural parameter space is
. Our basic strategy consists of linearizing the parameterized
convex system via splitting convex inequalities into linear ones by using the
Fenchel-Legendre conjugate. This approach yields that arbitrary bounded
right-hand side perturbations of the convex system turn on constant-by-blocks
perturbations in the linearized system. Based on advanced variational analysis,
we derive a precise formula for computing the exact Lipschitzian bound of the
feasible solution map of block-perturbed linear systems, which involves only
the system's data, and then show that this exact bound agrees with the
coderivative norm of the aforementioned mapping. In this way we extend to the
convex setting the results of [3] developed for arbitrary perturbations with no
block structure in the linear framework under the boundedness assumption on the
system's coefficients. The latter boundedness assumption is removed in this
paper when the decision space is reflexive. The last section provides the aimed
application to the convex case
Solving ill-posed bilevel programs
This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem an a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem
Proximal Analysis and the Minimal Time Function of a Class of Semilinear Control Systems
The minimal time function of a class of semilinear control systems is considered in Banach spaces, with the target set being a closed ball. It is shown that the minimal time functions of the Yosida approximation equations converge to the minimal time function of the semilinear control system. Complete characterization is established for the subdifferential of the minimal time function satisfying the Hamilton–Jacobi–Bellman equation. These results extend the theory of finite dimensional linear control systems to infinite dimensional semilinear control systems
The Radius of Metric Subregularity
There is a basic paradigm, called here the radius of well-posedness, which
quantifies the "distance" from a given well-posed problem to the set of
ill-posed problems of the same kind. In variational analysis, well-posedness is
often understood as a regularity property, which is usually employed to measure
the effect of perturbations and approximations of a problem on its solutions.
In this paper we focus on evaluating the radius of the property of metric
subregularity which, in contrast to its siblings, metric regularity, strong
regularity and strong subregularity, exhibits a more complicated behavior under
various perturbations. We consider three kinds of perturbations: by Lipschitz
continuous functions, by semismooth functions, and by smooth functions,
obtaining different expressions/bounds for the radius of subregularity, which
involve generalized derivatives of set-valued mappings. We also obtain
different expressions when using either Frobenius or Euclidean norm to measure
the radius. As an application, we evaluate the radius of subregularity of a
general constraint system. Examples illustrate the theoretical findings.Comment: 20 page
On a Convex Set with Nondifferentiable Metric Projection
A remarkable example of a nonempty closed convex set in the Euclidean plane
for which the directional derivative of the metric projection mapping fails to
exist was constructed by A. Shapiro. In this paper, we revisit and modify that
construction to obtain a convex set with smooth boundary which possesses the
same property
Set optimization - a rather short introduction
Recent developments in set optimization are surveyed and extended including
various set relations as well as fundamental constructions of a convex analysis
for set- and vector-valued functions, and duality for set optimization
problems. Extensive sections with bibliographical comments summarize the state
of the art. Applications to vector optimization and financial risk measures are
discussed along with algorithmic approaches to set optimization problems
Optimality conditions in convex multiobjective SIP
The purpose of this paper is to characterize the weak efficient solutions, the efficient solutions, and the isolated efficient solutions of a given vector optimization problem with finitely many convex objective functions and infinitely many convex constraints. To do this, we introduce new and already known data qualifications (conditions involving the constraints and/or the objectives) in order to get optimality conditions which are expressed in terms of either Karusk–Kuhn–Tucker multipliers or a new gap function associated with the given problem.This research was partially cosponsored by the Ministry of Economy and Competitiveness (MINECO) of Spain, and by the European Regional Development Fund (ERDF) of the European Commission, Project MTM2014-59179-C2-1-P
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