DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

ON A GENERAL MINIMIZATION PROBLEM
WITH CONSTRAINTS
VY K. LE AND DUMITRU MOTREANU
The paper studies the existence of solutions and necessary conditions of optimality for
a general minimization problem with constraints. We focus mainly on the case where
the cost functional is locally Lipschitz. Applications to an optimal control problem and
Lagrange multiplier rule are given.
Copyright © 2006 V. K. Le and D. Motreanu. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The paper deals with the following general minimization problem with constraints:
inf Φ(v).
v∈S
(P)
Here, Φ : X → R∪{+∞} is a function on a Banach space X and S is an arbitrary nonempty
subset of X. We suppose that S ∩ dom(Φ) ≠∅, where the notation dom(Φ) stands for
the effective domain of Φ, that is,
dom(Φ) = { x ∈ X : Φ(x) < +∞ } . (1.1)
First, we discuss the existence of solutions to problem (P). Precisely, we give an existence
result making use of a new type of Palais-Smale condition formulated in terms of
tangent cone to the set S and of contingent derivative for the function Φ.Asaparticular
case, one recovers the global minimization result for a locally Lipschitz functional satisfying
the Palais-Smale condition in the sense of Chang (cf. [7]). Then, by means of the
notion of generalized gradient (see Clarke [8]), we obtain necessary conditions of optimality
for problem (P) in the case where the cost functional Φ is locally Lipschitz. A
specific feature of our optimality conditions consists in the fact that the set of constraints
S is basically involved through its tangent cone. In addition, the costate variable provided
by the given necessary conditions makes use essentially of the imposed tangency hypothesis.
Finally, we present two applications of the necessary conditions of optimality that
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 645–654