Remarks on Lagrange Multiplier Rules in Set Valued Optimization

TL;DR

This paper compares three Lagrange multiplier rules in set valued optimization and introduces a generalized rule showing the equivalence of weak solutions to the constrained problem and the Lagrangian under mild assumptions.

Contribution

It provides a unifying generalization of existing Lagrange multiplier rules for set valued optimization problems, establishing their equivalence under mild conditions.

Findings

Unified framework for Lagrange multiplier rules

Equivalence of weak solutions to constrained and Lagrangian problems

Generalization applicable under mild assumptions

Abstract

In this note, three Lagrange multiplier rules introduced in the literature for set valued optimization problems are compared. A generalization of all three results is given which proves that under rather mild assumptions, $x$ is a weak solution to the constrained problem, if and only if it is a weak solution to the Lagrangian.