#### Title

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

#### Document Type

Technical Report

#### 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 loo(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.

#### Number in Series

2010.14

#### Disciplines

Applied Mathematics | Mathematics

#### AMS Subject Classification

90C34, 90C25, 49J52, 49J53, 65F22

#### Recommended Citation

Cánovas, M J.; Lopez, M A.; Mordukhovich, Boris S.; and Parra, J, "Quantitative Stability and Optimality Conditions in Convex Semi-Infinite and Infinite Programming" (2010). *Mathematics Research Reports*. 81.

http://digitalcommons.wayne.edu/math_reports/81

## Comments

This research was partially supported by grants MTM2008-06695-C03 (01-02) from MICINN (Spain); and by the US National Science Foundation under grants DMS-0603848 and DMS-1007132.