Optimization Problems with Nearly Convex Objective Functions and Nearly Convex Constraint Sets

TL;DR

This paper explores the relationship between nearly convex optimization problems and associated convex problems, establishing optimality conditions and Lagrange multiplier rules, with illustrations through concrete examples.

Contribution

It introduces a method to associate nearly convex problems with convex ones, deriving optimality and multiplier rules for the nearly convex setting.

Findings

Established relationships between nearly convex and convex problems.

Derived Fermat's optimality conditions for nearly convex problems.

Formulated Lagrange multiplier rules under constraints.

Abstract

To every nearly convex optimization problem, that is a minimization problem with a nearly convex objective function and a nearly convex constraint set, we associate a uniquely defined convex optimization problem with a lower semicontinuous objective function and a closed constraint set. Interesting relationships between the original nearly convex problem and the associated convex problem are established. Optimality conditions in the form of Fermat's rules are obtained for both problems. We then get a Lagrange multiplier rule for a nearly convex optimization problem under a geometrical constraint and functional constraints from the Kuhn-Tucker conditions for the associated convex optimization problem. The obtained results are illustrated by concrete examples.