Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality. We also introduce the concept of quasi weakly efficient solution for the above problems and divide all efficient solutions under consideration into the \emph {Pareto-type group} containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the \emph{weak Pareto-type group} containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. Formulated here necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting.


Nonlinear Analysis 75 (2012) 1305–1323