Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 21, No 3, (1995), 185p-200p
The author recently introduced a regularity assumption for derivatives
of set-valued mappings, in order to obtain first order necessary conditions of
optimality, in some generalized sense, for nondifferentiable control problems governed
by variational inequalities. It was noticed that this regularity assumption
can be viewed as a symmetry condition playing a role parallel to that of the wellknown
symmetry property of the Hessian of a function at a given point. In this
paper, we elaborate this point in a more detailed way and discuss some related
questions. The main issue of the paper is to show (using this symmetry condition)
that necessary conditions of optimality alluded above can be shown to be
also sufficient if a weak pseudo-convexity assumption is made for the subgradient
operator governing the control equation. Some examples of application to concrete
situations are presented involving obstacle problems.
* This work was completed while the author was visiting the University of Limoges. Support from the laboratoire “Analyse non-linéaire et Optimisation” is gratefully acknowledged.