Directional Compactness, Approximations and Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Constraints

In this article, we introduce and study a natural version of the directional compactness, which can be viewed as one of the effective tools for constructing sufficient conditions in nonsmooth vector equilibrium problems. We also provide the generalized Hadamard directional derivative notion which is closely related to a version of contingent derivative. The relation among the first-order approximations/the Clarke generalized Jacobian and the generalized Hadamard directional differentiability is formulated. Using the tool of approximations, a new version of the constraint qualification of the (CQ) type is proposed for establishing KT-type necessary nonsmooth optimality conditions via the generalized Hadamard directional derivatives for the weak/and strict efficiency of constrained nonsmooth vector equilibrium problems. As applications, we study a nonsmooth vector equilibrium problem with set, cone constraints using approximations and the constraint qualification (CQ).