Constraint Qualifications and Optimality Conditions for Nonconvex Semi-Infinite and Infinite Programs

TL;DR

This paper extends classical constraint qualifications to infinite and semi-infinite nonlinear programming problems, deriving new optimality conditions applicable in both finite and infinite-dimensional spaces.

Contribution

It introduces novel constraint qualification conditions for infinite and semi-infinite programs and derives first-order necessary optimality conditions in these settings.

Findings

Extended Mangasarian-Fromovitz and Farkas-Minkowski qualifications.

Computed normal cones for feasible sets using variational analysis.

Derived new first-order optimality conditions for complex programs.

Abstract

The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical Mangasarian-Fromovitz and Farkas-Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for efficient computing the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation. In the further development we derive first-order necessary optimality conditions for infinite and semi-infinite programs, which are new in both finite-dimensional and infinite-dimensional settings.