On strong second-order optimality conditions under relaxed constant rank constraint qualification

TL;DR

This paper establishes first- and second-order optimality conditions for nonlinear programming under a generalized constraint qualification, using only the inverse function theorem, making the proofs self-contained.

Contribution

It introduces a self-contained proof of optimality conditions under the relaxed constant rank constraint qualification, extending the theoretical framework with minimal prerequisites.

Findings

Optimality conditions are derived using only the inverse function theorem.

The relaxed constant rank constraint qualification generalizes LICQ.

Proofs are simplified and self-contained.

Abstract

We discuss the (first- and second-order) optimality conditions for nonlinear programming under the relaxed constant rank constraint qualification. This condition generalizes the so-called linear independence constraint qualification. Although the optimality conditions are well established in the literature, the proofs presented here are based solely on the well-known inverse function theorem. This is the only prerequisite from real analysis used to establish two auxiliary results needed to prove the optimality conditions, thereby making this paper totally self-contained.