On the Necessity of the Sufficient Conditions in Cone-Constrained Vector Optimization

TL;DR

This paper explores the conditions under which cone-constrained vector optimization problems are guaranteed to have global solutions, generalizing classical scalar and Pareto optimality results to vector cases with cone constraints.

Contribution

It introduces the notion of Fritz John pseudoinvexity for cone-constrained problems and establishes equivalences with global optimality, extending existing scalar and Pareto optimality theories.

Findings

Fritz John pseudoinvexity characterizes global solutions in cone-constrained vector problems.

Pseudoconvexity of quasiconvex vector functions is equivalent to all critical points being global minimizers.

Results generalize scalar and Pareto optimality conditions to vector optimization with cone constraints.

Abstract

The object of investigation in this paper are vector nonlinear programming problems with cone constraints. We introduce the notion of a Fritz John pseudoinvex cone-constrained vector problem. We prove that a problem with cone constraints is Fritz John pseudoinvex if and only if every vector critical point of Fritz John type is a weak global minimizer. Thus, we generalize several results, where the Paretian case have been studied. We also introduce a new Frechet differentiable pseudoconvex problem. We derive that a problem with quasiconvex vector-valued data is pseudoconvex if and only if every Fritz John vector critical point is a weakly efficient global solution. Thus, we generalize a lot of previous optimality conditions, concerning the scalar case and the multiobjective Paretian one. Additionally, we prove that a quasiconvex vector-valued function is pseudoconvex with respect to the same cone if and only if every vector critical point of the function is a weak global minimizer, a result, which is a natural extension of a known characterization of pseudoconvex scalar functions.