#### Document Type

Technical Report

#### Abstract

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 is finite, this space is intrinsically infinite-dimensional (nonreflexive and nonseparable) of the l00-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.

#### Number in Series

2009.06

#### Disciplines

Applied Mathematics | Mathematics | Numerical Analysis and Computation

#### AMS Subject Classification

90C34, 90C05, 49J52, 49J53, 65F22

#### Recommended Citation

Cánovas, M J.; Lopez, M A.; Mordukhovich, Boris S.; and Parra, J, "Variational Analysis in Semi-Infinite and Infinite Programming, I: Stability of Linear Inequality Systems of Feasible Solutions" (2009). *Mathematics Research Reports*. 67.

http://digitalcommons.wayne.edu/math_reports/67

## Comments

This research was partially supported by grants MTM2005-08572-C03 {01-02) from MEC (Spain) and FEDER (EU), and MTM2008-06695-C03 (01-02) from MICINN (Spain); and by the National Science Foundation (USA) under grant DMS-0603846.