On composite vector variational-like inequalities and vector optimization problems

In this paper, we introduce a class of (weak) composite vector variational-like inequality problems and establish its relationship with composite vector optimization problem. We also prove the relation of a vector critical point in composite vector optimization problem with its weak efficient point, under the assumption of composite pseudo invexity. Using KKM Lemma, we derive result for existence of solutions of composite vector variational-like inequality problem. Furthermore, we define a gap function for the composite vector variational-like inequality problem and finally, as an application, we study a system of composite vector optimization problems and system of vector variational-like inequality problems, whose solutions imply the solution of Nash equilibrium problem. Examples are provided to illustrate the derived results. © 2015, Springer-Verlag Berlin Heidelberg.