Optimization Problems with Nearly Convex Objective Functions and Nearly Convex Constraint Sets
Nguyen Nang Thieu, Nguyen Dong Yen
TL;DR
The paper addresses optimization problems with nearly convex objectives and nearly convex constraint sets by associating each problem with a convex problem whose objective is the lower semicontinuous hull $\bar f$ on the closed feasible set $\bar D$, enabling a direct transfer of optimality conditions under a regularity condition on relative interiors. It establishes Fermat’s rule for both problems and derives a Lagrange multiplier/Kuhn–Tucker framework for the nearly convex problem via the convex counterpart, supplemented by concrete examples and a comparison of two near-convexity notions. The key contributions include a systematic method to obtain optimality conditions for nearly convex problems, the use of the lower semicontinuous hull to connect to convex analysis, and a detailed examination of geometric and functional constraint settings (including a generalized Slater condition). The results enhance understanding of near-convex optimization and provide practical tools for applying convex analysis techniques to broader nonconvex settings.
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.