Existence Results and KKT Optimality Conditions for Generalized Quasiconvex Functions

TL;DR

This paper introduces $e$-quasiconvex functions, extending convexity concepts, and establishes conditions for solution set properties and optimality criteria in nonconvex optimization problems.

Contribution

It defines $e$-quasiconvexity, extends key properties of quasiconvex functions, and proves KKT conditions are sufficient for optimality in this broader context.

Findings

Solution sets are nonempty and compact under certain conditions.

KKT conditions are sufficient for optimality with $e$-quasiconvex constraints.

Examples demonstrate applicability to non-quasiconvex problems.

Abstract

We studied a new notion of generalized convex functions called $e$-quasi\-con\-ve\-xi\-ty, which encompasses both quasiconvex and $e$-convex functions, including all Lipschitz functions. By extending the standard properties of quasiconvex functions to $e$-quasiconvex functions, we establish sufficient conditions for the nonemptiness and compactness of the solution set when minimizing an $e$-quasiconvex function, leveraging generalized asymptotic functions, a result which remains applicable even when the set of minimizers is nonconvex. Furthermore, in the differentiable case, we ensure the sufficiency of the KKT optimality conditions when the constraint functions in the mathematical programming problems are $e$-quasiconvex. Finally, we illustrate our new results with several nonconvex (non-quasiconvex) examples.