Abadie condition for infinite programming problems under Relaxed Constant Rank Constraint Qualification Plus

TL;DR

This paper establishes an Abadie condition for infinite programming problems using a new Relaxed Constant Rank Constraint Qualification Plus in Banach spaces, enabling analysis of Lagrange multipliers.

Contribution

It introduces a novel constraint qualification for infinite problems and proves the Abadie condition under this new framework.

Findings

Proves the Abadie condition under the Relaxed Constant Rank Constraint Qualification Plus.

Discusses the existence of Lagrange multipliers in infinite-dimensional settings.

Utilizes Rank and Ljusternik Theorems for the analysis.

Abstract

We consider infinite programming problems with constraint sets defined by systems of infinite number of inequalities and equations given by continuously differentiable functions defined on Banach spaces. In the approach proposed here we represent these systems with the help of coefficients in a given Schauder basis. We prove the Abadie condition under the new infinite-dimensional Relaxed Constant Rank Constraint Qualification Plus and we discuss the existence of Lagrange multipliers. The main tools are: Rank Theorem and Ljusternik Theorem.