Nonempty Boundedness Of Solution Sets For Generalized Convex Function Minimum Problems

Optimization theory is widely used in many fields such as economic planning,government decision-making,production management,transportation,military and national defense,engineering technology and management science.The existence of solutions of function minimum problem is a basic problem in optimization theory.The boundedness of solution set is very important in the design of numerical algorithm.Convexity plays an important role in optimization theory.When dealing with practical problems,the functions encountered are often non-convex.In this thesis,we study the nonemptiness and boundedness of the solution set of the quasiconvex function minimization problems,and of the pseudoconvex function minimization problems,and of the weakly efficient solution sets for pseudoconvex vector optimization problems.The detailed arrangement of the thesis is as follows:In chapter 1,we introduce the research background and research status of the minimum problems of pseudoconvex functions and quasiconvex functions the development of exceptional family of elements.Moreover,we introduce the common symbols,basic concepts and lemmas used in this dissertation.In chapter 2,we investigate the nonemptiness and boundedness of the solution set for minimum problems of quasiconvex functions.We utilize asymptotic analysis to deal with the quasiconvex optimization problems in reflexive Banach spaces.The definition of q-asymptotic function of quasiconvex function on the nonempty closed convex set is presented.The characterization of the existence of the solution set of the quasiconvex function minimum problems is established by using the q-asymptotic function of the quasiconvex function on the nonempty closed convex set.The results of this paper extend the classical results for nonemptiness and boundness of the solution set of convex function minimum problems to quasiconvex function minimum problems.We establish a sufficient condition for the nonemptiness and boundness of the solution setof equilibrium problems.In chapter 3,we investigate the nonemptiness and boundedness of solution set for minimum problems of pseudoconvex functions in reflexive Banach Spaces by using the exceptional family of elements.Firstly,we give the definition of the exceptional family of elements for minimum problems of pseudoconvex functions,we show that the nonexistence of an exceptional family of elements is a necessary condition for the solvability of minimum problems of pseudoconvex functions.We obtain some equivalent conditions for the minimum problems of pseudoconvex functions to be nonempty bounded set.We apply this result to the quadratic fractional programming problem.The existence of solutions of quadratic fractional programming problem is obtained by using the exceptional family of elements.In chapter 4,we investigate the nonemptiness and boundedness of weakly efficient solution sets for pseudoconvex vector optimization problems in reflexive Banach Spaces.We prove that the nonemptiness and boundedness of the solution of a pseudoconvex vector optimization problem is equivalent to the nonemptiness and boundedness of the solution of any scalar pseudoconvex optimization problem and the nonemptiness and boundedness of the solution set of a scalar optimization problem with m component functions.And we give a sufficient and necessary condition,which ensure the weakly efficient solution set of the pseudoconvex vector optimization problem is nonempty and bounded.