Journal of Optimization Theory and Applications

, Volume 181, Issue 1, pp 144–162

# Characterizations of Solution Sets of Differentiable Quasiconvex Programming Problems

• Vsevolod I. Ivanov
Article

## Abstract

In this paper, we study some problems with a continuously differentiable and 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 program with a continuously differentiable and quasiconvex objective function, 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.

## Keywords

Quasiconvex function Characterizations of the solution set Quasiconvex program Pseudoconvex function KKT Conditions

## Mathematics Subject Classification

90C26 26B25 90C46

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