Optimality conditions for multiobjective problems

Optimality conditions for multiobjective problems

Federal University of Paraná (January 2016 – December 2016)

A study was carried out on the existence of solutions and on the necessary and sufficient optimality conditions for nonlinear programming problems. The proofs of the Fritz–John and Karush–Kuhn–Tucker conditions were based on the approach via Gordan-type alternative theorems. We saw that the KKT conditions are sufficient for optimality when the functions are convex, and necessary whenever a suitable constraint-qualification holds; from these conditions we aim to extend the results already obtained to multiobjective programming problems. Additionally, we examined topics in generalized convexity—namely the concepts of invexity and KT-invexity. If all functions are invex, the sufficiency of the KKT conditions remains valid. However, there exist nonlinear programming problems whose KKT points are optimal even though the problem is not invex. KT-invexity, a variation on invexity, defines the largest class of problems for which the KKT conditions are both necessary and sufficient for optimality.

Doctoral Members: Lucelina Batista dos Santos (PI)

Students: Everton Silva