On Vector Equilibria, Vector Optimization and Vector Variational Inequalities

It is well-known that, under certain conditions, network equilibrium, optimization and variational inequality problems are equivalent. Hence, solution algorithms to solve any of the three problems can be used to solve the other problems. Vector network equilibrium problems lead to analogous definitions of vector optimization (VOP) and vector variational inequality (VVI) problems. Investigating whether a similar equivalence exists in the vector valued case suggests itself, in particular to derive solution algorithms for vector equilibrium problems (VEQ). Unfortunately, the three problems are no longer equivalent in the vector valued case. We show under which assumptions a solution of VOP solves VEQ. Even though a solution of VVI is a solution of VEQ, the converse is not true. We demonstrate structural properties of solutions of VEQ that prevent them from being solutions of VVI and show under which assumptions VVI and VEQ are equivalent. We also comment in more detail on some results within the literature related to concepts of vector equilibria.