Connectedness Of Solution Sets For Vector Equilibrium Problems

Vector equilibrium problem is a wide class of mathematical model,including vector variational inequality problem and vector optimization problem.It is widely applied in the fields of economy and finance,transportation,resource allocation and engineering management.This dissertation mainly studies the connectedness of solution sets for vector equilibrium problem and it's related problems.This paper is arranged as follows:In Chapter 1,we briefly introduce the research background and status of vector equilibrium problem and it's related problems,and give some basic symbols,concepts and lemmas to be used later.In Chapter 2,by employing the connectedness of dual cone,we prove that the nonempti-ness and boundedness of the sets of weak efficient solutions for convex vector optimization problems are equivalent to the nonemptiness and boundedness of solution sets for each scalar optimization problem in reflexive Banach spaces.We give a sufficient condition of the nonemptiness and boundedness of the sets of strong efficient solutions for convex vector optimization problems.By employing the weak compactness of the sets of weak efficient solutions of convex vector optimization problems,we investigate the weak connectedness of the sets of weak efficient solutions and strong efficient solutions of convex vector optimization problems.When each component function is strictly convex,we investigate weak path-connectedness of the sets of strong efficient solutions for convex vector optimization problems.In Chapter 3,by scalarization,we investigate the connectedness of the sets of weak efficient solutions for convex vector optimization problems on unbounded closed convex sets in finite dimensional spaces.When the vector-valued function is cone lower semicontinuous and cone convex,we prove that the solution set mapping is upper semicontinuous,and investigate the connectedness of the solution set by using the connectedness of compact convex base of polar negative cone.When the vector-valued function is cone semicontinuous and cone strictly convex,we investigate the path-connectedness of the sets of weak efficient solutions for convex vector optimization problem,and the compactness of the sets of weak efficient solutions for composite multiobjective program problems,and the connectedness of the sets of weak efficient solutions for composite multiobjective program problems,and the connectedness of the solution sets for affine vector variational inequality problems.In Chapter 4,we establish a nonconvex separation theorem,that is,a neither convex nor closed set and a noncompact set can be separated.We prove that the solution set of the strong vector equilibrium problem can be expressed as the union of the solution sets of a series of nonlinear scalar problems.By using the nonconvex separation theorem and the coercivity condition,we investigate the connectedness of the solution sets for strong vector equilibrium problems on the closed convex set.