#### Document Type

Technical Report

#### 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.

#### Number in Series

2011.06

#### Disciplines

Applied Mathematics | Mathematics

#### Recommended Citation

Mordukhovich, Boris S. and Nghia, T T. A., "Constraint Qualifications and Optimality Conditions for Nonconvex Semi-Infinite and Infinite Programs" (2011). *Mathematics Research Reports*. 87.

https://digitalcommons.wayne.edu/math_reports/87

## Comments

Dedicated to Jon Borwein in honor of his 60th birthday. This research was partially supported by the USA National Science Foundation under grants DMS-0603846 and DMS-1007132 and by the Australian Research Council under grant DP-12092508.