Weak lower subdifferentials and applications

Abstract

We introduce the class of weakly lower subdifferentiable functionals, by relaxing the notion of lower subdifferentiability in the sense of Plastria, thus obtaining a local notion for strictly quasiconvex functionals. We state its relations with the lower DiNi derivatives and prove that, for Gâteaux differentiate functionals, weak lower subdifferentiability is equivalent to pseudoconvexity. After giving some simple calculus rules for weak lower subdifferentials, we obtain a Kuhn-Ttjckeb type theorem for quasiconvex optimization problems in terms of weak lower subgradients. Finally, we study the weak lower subdifferentiability of the minimal time function of linear control systems.