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 $J$. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is $l_{\infty}(J)$. 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

Quantitative Stability and Optimality Conditions in Convex Semi-Infinite and Infinite Programming

This paper concerns parameterized convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional Banach (resp. finite-dimensional) spaces and that are indexed by an arbitrary fixed set T . Parameter perturbations on the right-hand side of the inequalities are measurable and bounded, and thus the natural parameter space is $l_{\infty}(T)$. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map, which involves only the system data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. On one hand, in this way we extend to the convex setting the results of [4] developed in the linear framework under the boundedness assumption on the system coefficients. On the other hand, in the case when the decision space is reflexive, we succeed to remove this boundedness assumption in the general convex case, establishing therefore results new even for linear infinite and semi-infinite systems. The last part of the paper provides verifiable necessary optimality conditions for infinite and semi-infinite programs with convex inequality constraints and general nonsmooth and nonconvex objectives. In this way we extend the corresponding results of [5] obtained for programs with linear infinite inequality constraints

On the stability of the optimal value and the optimal set in optimization problems

The paper develops a stability theory for the optimal value and the optimal set mapping of optimization problems posed in a Banach space. The problems considered in this paper have an arbitrary number of inequality constraints involving lower semicontinuous (not necessarily convex) functions and one closed abstract constraint set. The considered perturbations lead to problems of the same type as the nominal one (with the same space of variables and the same number of constraints), where the abstract constraint set can also be perturbed. The spaces of functions involved in the problems (objective and constraints) are equipped with the metric of the uniform convergence on the bounded sets, meanwhile in the space of closed sets we consider, coherently, the Attouch-Wets topology. The paper examines, in a unified way, the lower and upper semicontinuity of the optimal value function, and the closedness, lower and upper semicontinuity (in the sense of Berge) of the optimal set mapping. This paper can be seen as a second part of the stability theory presented in [17], where we studied the stability of the feasible set mapping (completed here with the analysis of the Lipschitz-like property)

Robust Stability and Optimality Conditions for Parametric Infinite and Semi-Infinite Programs

This paper primarily concerns the study of parametric problems of infinite and semi-infinite programming, where functional constraints are given by systems of infinitely many linear inequalities indexed by an arbitrary set T, where decision variables run over Banach (infinite programming) or finite-dimensional (semi-infinite case) spaces, and where objectives are generally described by nonsmooth and nonconvex cost functions. The parameter space of admissible perturbations in such problems is formed by all bounded functions on T equipped with the standard supremum norm. Unless the index set T is finite, this space is intrinsically infinite-dimensional (nonreflexive and nonseparable) of the l(infinity)-type. By using advanced tools of variational analysis and generalized differentiation and largely exploiting underlying specific features of linear infinite constraints, we establish complete characterizations of robust Lipschitzian stability (with computing the exact bound of Lipschitzian moduli) for parametric maps of feasible solutions governed by linear infinite inequality systems and then derive verifiable necessary optimality conditions for the infinite and semi-infinite programs under consideration expressed in terms of their initial data. A crucial part of our analysis addresses the precise computation of coderivatives and their norms for infinite systems of parametric linear inequalities in general Banach spaces of decision variables. The results obtained are new in both frameworks of infinite and semi-infinite programming

On calmness of the optimal value function

The paper is devoted to the calmness from below/from above for the optimal value function of parametric optimization problems, where we are specifically interested in perturbed semi-infinite programs. A main intention is to revisit classical results and to derive refinements of them. In particular, we show in the context of semi-infinite optimization that calmness from below for j holds under quasiconvexity of the data functions and compactness of the solution set, which extends results on the lower semicontinuity of Illustrative examples are given, which demonstrate the significance of the imposed assumptions even in the case of linear and quadratic programs

Calmness modulus of linear semi-infinite programs

Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of the argmin mapping and the same property for the (sub)level set mapping (with respect to the objective function), for semi-infinite programs and without requiring the uniqueness of the nominal solution, is explored, too, providing an upper bound on the calmness modulus of the argmin mapping. When confined to finitely constrained problems, we also provide a computable upper bound as it only relies on the nominal data and parameters, not involving elements in a neighborhood. Illustrative examples are provided.This research has been partially supported by grants MTM2011-29064-C03 (02-03) from MINECO, Spain, ACOMP/2013/062 from Generalitat Valenciana, Spain, grant C10E08 from ECOS-SUD, and grant DP110102011 from the Australian Research Council

Subdifferentials and Stability Analysis of Feasible Set and Pareto Front Mappings in Linear Multiobjective Optimization

The paper concerns multiobjective linear optimization problems in R^n that are parameterized with respect to the right-hand side perturbations of inequality constraints. Our focus is on measuring the variation of the feasible set and the Pareto front mappings around a nominal element while paying attention to some specific directions. This idea is formalized by means of the so-called epigraphical multifunction, which is defined by adding a fixed cone to the images of the original mapping. Through the epigraphical feasible and Pareto front mappings we describe the corresponding vector subdifferentials, and employ them to verifying Lipschitzian stability of the perturbed mappings with computing the associated Lipschitz moduli. The particular case of ordinary linear programs is analyzed, where we show that the subdifferentials of both multifunctions are proportional subsets. We also provide a method for computing the optimal value of linear programs without knowing any optimal solution. Some illustrative examples are also given in the paper.Comment: 23 page

Variational Analysis in Semi-Infinite and Infinite Programming, I: Stability of Linear Inequality Systems of Feasible Solutions

This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to parametric problems of semi-infinite and infinite programming, where decision variables run over finite-dimensional and infinite-dimensional spaces, respectively. Part I is primarily devoted to the study of robust Lipschitzian stability of feasible solutions maps for such problems described by parameterized systems of infinitely many linear inequalities in Banach spaces of decision variables indexed by an arbitrary set T. The parameter space of admissible perturbations under consideration is formed by all bounded functions on T equipped with the standard supremum norm. Unless the index set T is finite, this space is intrinsically infinite-dimensional (nonreflexive and nonseparable) of the l∞ type. By using advanced tools of variational analysis and exploiting specific features of linear infinite systems, we establish complete characterizations of robust Lipschitzian stability entirely via their initial data with computing the exact bound of Lipschitzian moduli. A crucial part of our analysis addresses the precise computation of the coderivative of the feasible set mapping and its norm. The results obtained are new in both semi-infinite and infinite frameworks. (A correction to the this article has been appended at the end of the pdf file.)This research was partially supported by grants MTM2005-08572-C03 (01-02) from MEC (Spain) and FEDER (EU), MTM2008-06695-C03 (01-02) from MICINN (Spain), and ACOMP/2009/047&133 from Generalitat Valenciana (Spain); National Science Foundation (USA) under grant DMS-0603846