On the multiplier rules

TL;DR

This paper presents new first-order necessary conditions for optimality in finite-dimensional constrained problems, weakening previous assumptions on continuity and differentiability, and providing formulations in the style of John's and KKT's theorems.

Contribution

It introduces generalized optimality conditions with relaxed assumptions, expanding the applicability of classical theorems to less smooth problems.

Findings

Established new first-order necessary conditions for constrained optimization.

Weakened assumptions of continuity and differentiability compared to classical results.

Provided formulations in the style of John's and KKT's theorems.

Abstract

We establish new results of first-order necessary conditions of optimality for finite-dimensional problems with inequality constraints and for problems with equality and inequality constraints, in the form of John's theorem and in the form of Karush-Kuhn-Tucker's theorem. In comparison with existing results we weaken assumptions of continuity and of differentiability.