First and Second Order Necessary and Sufficient Optimality Conditions of Fritz John Type for Vector Problems over Cones

TL;DR

This paper presents new proofs of Fritz John necessary and second-order optimality conditions for vector optimization problems over cones, utilizing fixed point theorems and directional derivatives, and discusses sufficiency for efficiency.

Contribution

It introduces novel proofs for Fritz John optimality conditions and extends them to second-order cases, also providing sufficiency criteria for efficiency in vector problems.

Findings

New proof of Fritz John necessary conditions using fixed point theorem

Extension to second-order Fritz John optimality conditions

Sufficient conditions for weak global and local efficiency

Abstract

In this paper, we obtain a new proof of Fritz John necessary optimality conditions for vector problems applying Kakutani fixed point theorem and Hadamard directional derivative. We also derive a similar proof of second-order Fritz John necessary optimality conditions. Sufficient conditions for weak global efficiency with generalized convex functions and local efficiency are provided.