Some Properties Of Solutions And Optimality Conditions For Vector Equilibrium Problems

Vector equilibrium problems, which is a natural extension of vector optimization and vector variational inequality, compose an important part of operations research. It contains the vector optimization, vector variational inequalities, vector Nash equilib-rium and vector complementarity problem as a special case. And it has been widely applied in mathematical programming, management science, engineering, mathemati-cal economics and social economic system and many other fields.Both scalarization and optimality conditions for vector equilibrium problems are important components in the vector equilibrium theory. They arc important foundation for developing algorithms. Moreover, mathematical models are generally approxima-tion of practical problems and solutions obtained by the iterative algorithm are gener-ally approximate solution. In particular, efficient solution set (or weak efficient solution set) may often be the empty set when the feasible set is noncompact. But the set of approximate solution in very weak conditions can be not empty. Therefore, studying approximate solutions to vector equilibrium problem has theoretical value and practi-cal significance. This article first introduce the concepts of ε-efficient solution, ε-weak efficient solution, ε-Henig efficient solutions and ε-global efficient solution to vec-tor equilibrium problem with constraints, and studies some properties of approximate efficient solutions and approximate weak efficient solutions. The relations between ap-proximate solutions and efficient solution are also discussed. Then, in Banach space, we give results on Chebyshev scalarization of e-efficient solutions, ε-weak efficient so-lutions, ε-Henig efficient solutions and ε-global efficient solutions in the application of generalized norm and also give the nonlinear scalarization results of vector equilibrium problem with constraints through nonlinear scalarization. Secondly, we present suffi-cient and necessary conditions for ε-efficient solutions, ε-weak efficient solutions, ε-Henig efficient solutions and e-global efficient solutions to vector equilibrium problem with constraints under objective function is cone-subconvexlike. Finally, we obtain the Kuhn-Tucker optimality conditions for vector equilibrium problem with constrained conditions under objective functions are cone-arcwise connected maps.