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

TL;DR

This paper develops precise stability bounds and optimality conditions for convex semi-infinite and infinite programming problems, extending previous linear results to more general convex settings and removing boundedness assumptions.

Contribution

It derives an exact Lipschitzian bound for the feasible solution map using variational analysis, extending linear results to convex systems and removing boundedness constraints in reflexive spaces.

Findings

Exact Lipschitzian bounds computed from system data

Extension of stability results to convex infinite systems

Verifiable necessary optimality conditions for complex programs

Abstract

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.