Abstract
In this paper, we introduce a new kind of approximate weakly efficient solutions to the set-valued vector equilibrium problems with constraints in locally convex Hausdorff topological vector spaces; then we discuss a relationship between the weakly efficient solutions and approximate weakly efficient solutions. Under the assumption of near cone-subconvexlikeness, by using the separation theorem for convex sets we establish Kuhn–Tucker-type and Lagrange-type optimality conditions for set-valued vector equilibrium problems, respectively.
Highlights
Vector optimization problems, vector variational inequality problems, vector complementarity problems, and vector saddle point problems are particular cases of vector equilibrium problems
Motivated by works in [3, 12, 15], in this paper, we introduce a new kind of approximate weakly efficient solutions to the set-valued vector equilibrium problems and reveal the relationship between weakly efficient solutions and approximate weakly efficient solutions
3 Approximate weakly efficient solutions Firstly, we introduce approximate weakly efficient solutions to the set-valued vector equilibrium problems with constraints
Summary
Vector variational inequality problems, vector complementarity problems, and vector saddle point problems are particular cases of vector equilibrium problems. 4, we establish Kuhn–Tucker-type sufficient and necessary optimality conditions for approximate weakly efficient solutions to the set-valued vector equilibrium problems. 5, we establish Lagrange-type sufficient and necessary optimality conditions for approximate weakly efficient solutions to the set-valued vector equilibrium problems. Definition 2.4 A vector x ∈ X0 is said to be a weakly efficient solution of (USOP)Tif there exists y ∈ F(x) such that (L(X0, T ) –y) ∩(– int C) = ∅, where L(X0, T ) = x∈X0 L(x, T ). We discuss the relationship between the approximate weakly efficient solutions and weakly efficient solutions to the set-valued vector equilibrium problems with constraints.
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