A Note on the Guignard Constraint Qualification and the Guignard Regularity Condition in Vector Optimization

Giorgio Giorgi

doi:10.4236/am.2013.44101

Abstract

Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions in differentiable vector optimization problems.

Highlights

In discussing a gap between multiobjective optimization and scalar optimization, Aghezzaf and Hachimi [2] state that “in multiobjective optimization problems, many authors have derived the first-order and second-order necessary conditions under the Abadie constraint qualification, but never under the Guignard constraint qualification”
Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions in differentiable vector optimization problems
We have described in the Introduction the “gap” occurring between scalar and vector optimization problems, generated by the use of the classical Guignard-GouldTolle constraint qualification

Summary

Introduction

In discussing a gap between multiobjective optimization and scalar optimization (a gap first pointed out by Wang and Yang [1]), Aghezzaf and Hachimi [2] state that “in multiobjective optimization problems, many authors have derived the first-order and second-order necessary conditions under the Abadie constraint qualification, but never under the Guignard constraint qualification”.

First Order Necessary Conditions

Again on the “Gap” between Scalar Problems and Vector Problems

Conclusions