Normality and uniqueness of Lagrange multipliers

Karla L Cortez, ,Javier F Rosenblueth

doi:10.3934/dcds.2018138

Abstract

In this paper we study, for certain problems in the calculus of variations and optimal control, two different questions related to uniqueness of multipliers appearing in first order necessary conditions. One deals with conditions under which a given multiplier associated with an extremal of a fixed function is unique, a property which, in nonlinear programming, is known to be equivalent to the strict Mangasarian-Fromovitz constraint qualification. We show that, for isoperimetric problems in the calculus of variations, a similar characterization holds, but not in optimal control where the corresponding condition is only sufficient for the uniqueness of the multiplier. The other question is related to the set of multipliers associated with all functions for which a solution to the constrained problem is given. We prove that, for both types of problems, this set is a singleton if and only if a strong normality assumption holds.

Highlights

In this paper we study, for certain classes of constrained optimization problems, uniqueness of Lagrange multipliers satisfying first order necessary conditions
To explain the relation between these two aspects let us first mention that, according to [12], Bazaraa, Sherali and Shetty [2] and Ben-Tal [5] provide a false result on second order necessary conditions and a correct result was derived by Kyparisis [15]
Following [12], the authors in [2, 5] claim that, assuming the linear independence constraint qualification, second order conditions hold on the set of tangential constraints relative to the original set of inequality and equality constraints

Summary

Introduction

In this paper we study, for certain classes of constrained optimization problems, uniqueness of Lagrange multipliers satisfying first order necessary conditions. There, for the problem of minimizing a given real-valued function on Rn subject to inequality and equality constraints, one can find one result on uniqueness of Lagrange multipliers (Theorem 3.10.4). A second result given in [15], connecting the strict Mangasarian-Fromovitz constraint qualification with second order necessary conditions, is stated as follows.

Results

Conclusion