Abstract
In locally convex Hausdorff topological vector spaces, the approximate Benson efficient solution is proposed for set-valued equilibrium problems and its relationship to the Benson efficient solution is discussed. Under the assumption of generalized convexity, by using a separation theorem for convex sets, Kuhn–Tucker-type and Lagrange-type optimality conditions for set-valued equilibrium problems are established, respectively.
Highlights
The vector equilibrium problem is a broad problem in many practical fields. It covers many typical mathematical problems, for instance, vector optimization, variational inequality, vector Nash equilibrium, vector complementarity, and so on
1 Introduction The vector equilibrium problem is a broad problem in many practical fields
Because of the universality and unity of the problems involved and the profundity of solving them, vector equilibrium has become a hot issue in the field of nonlinear analysis and operational research [1,2,3,4,5,6]
Summary
The vector equilibrium problem is a broad problem in many practical fields. It covers many typical mathematical problems, for instance, vector optimization, variational inequality, vector Nash equilibrium, vector complementarity, and so on. Definition 2.3 A vector x ∈ Ω is called a Benson efficient solution to (Υ -SEPC) if clcone Υ (x, Ω) + S ∩ (–S) = {0Y }. Definition 2.5 ([15]) A vector x ∈ E is called a Benson efficient solution to (SOP) if there exists y ∈ F(x) such that clcone F(E) – y + S ∩ (–S) = {0Y }.
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