Characterizations of Solution Sets of Differentiable Quasiconvex Programming Problems

• Published in: Journal of optimization theory and applications Vol. 181; no. 1; pp. 144 - 162
• Main Author: Ivanov, Vsevolod I.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-04-2019; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Economic models; Engineering; Lagrange multiplier; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Out of stock; Theory of Computation; Quasiconvex program; Pseudoconvex function; 90C46; Characterizations of the solution set; KKT Conditions; 90C26; Quasiconvex function; 26B25
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-018-1379-1

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.