Characterizations of Solution Sets of Fr\'echet Differentiable Problems with Quasiconvex Objective Function

TL;DR

This paper characterizes the solution sets of quasiconvex, continuously differentiable optimization problems, revealing conditions on gradients and providing Lagrange multiplier descriptions, with implications for constrained optimization.

Contribution

It introduces a novel dichotomy for the gradient behavior over solution sets and offers new characterizations and Lagrange multiplier conditions for quasiconvex problems.

Findings

Either the gradient is non-zero and constant over the solution set or zero everywhere on it.

Provides solution set characterizations when a solution is known.

Derives Lagrange multiplier conditions for quasiconvex constrained problems.

Abstract

In this paper, we study some problems with continuously differentiable quasiconvex objective function. We prove that exactly one of the following two alternatives holds: (I) the gradient of the objective function is different from zero over the solution set and the normalized gradient is constant over it; (II) the gradient of the objective function is equal to zero over the solution set. As a consequence, we obtain characterizations of the solution set of a quasiconvex continuously differentiable program, provided that one of the solutions is known. We also derive Lagrange multiplier characterizations of the solutions set of an inequality constrained problem with continuously differentiable objective function and differentiable constraints, which are all quasiconvex on some convex set, not necessarily open. We compare our results with the previous ones. Several examples are provided.