Uniqueness Of Multipliers Research Articles

Normality and Uniqueness of Multipliers in Isoperimetric Control Problems

In this paper, we introduce the notion of normality relative to a set of constraints in isoperimetric control problems and study its relationship with the classic notion of normality, as well as the existence and uniqueness of Lagrange multipliers satisfying the maximum principle. We show that this notion leads to characterizing the uniqueness of a given multiplier, which also turns out to be equivalent to a strict Mangasarian–Fromovitz condition (as in the finite-dimensional case). Finally, we show that, if the cost function is allowed to vary between those for which a solution to the constrained problem is given, then the set of multipliers associated with each of them is a singleton, if and only if a strong normality assumption holds.

Normality and uniqueness of Lagrange multipliers

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.