Study On Properly Efficient Solutions And Approximate Solutions For Multiobjective Programming Problems

Multiobjective programming problem is the extension of numerical value opti-mization problem, its all results can apply to numerical value optimization problem. Multiobjective programming is an interdiscipline between the applied mathematics and decision science, the theory of this topic involves many disciplines, such as:convex analysis, nonsmooth analysis, nonlinear analysis, variational analysis, and so on. at the same time, the theory and methods of Multiobjective programming are widely applied in many areas, such as:economic planning, financial investment, engineering design, environmental protection, health and medical community, transportation, etc. Therefore, the research on this topic has both theoretical value and practical signifi-cance. This whole thesis is divided into four chapters, we mainly study the theory of multiobjective programming in two aspects:optimality conditions of properly efficient solutions and nonlinear scalarizations of approximate solutions for multiobjective pro-gramming. The main results, obtained in this dissertation, may be summarized as follows:1. In chapter 1, we give a brief introduction about research significance and contents of multiobjective programming problems. And we sum up the development history and current situation of multiobjective programming as well as three aspects:properly efficient solution, approximate solution and scalarization related to this article. Finally, we state the main contents studied in this thesis.2. Chapter 2 is committed to study the optimality conditions of Proximal properly efficient solutions for multiobjective programming problems. First, linear scalarization for proximal properly efficient points to a closed set is presented under generalized convexity assumption, and linear scalarization of Proximal properly efficient solutions for multiobjective programming problems is obtained by applying the result to the problems. Second, fuzzy optimality condition for Proximal properly efficient solutions is obtained by using Proximal subdifferential. Finally, the equivalency among the Proximal proper efficiency, Benson proper efficiency and Borwein proper efficiency in multiobjective programming problems is discussed. The connectedness of Proximal proper effi(?) solutions sets for set-valued optimization is considered under upper semi-continuous and generalized convexity assumptions.3. Chapter 3 aims at nonlinear scalarizations of approximate solutions for multiob-jective programming. First, we give some sufficient conditions for these approximate solu-tions via nonlinear scalarization without any convexity assumptions. Our results correct the mistakes of several existence results, and some examples are given to illustrated the main results. Furthermore, the nonlinear scalarizations of approximate efficient solutions and approximate properly efficient solutions for multiobjective programming problems are obtained by utilizing norm to establish scalar problems. Moreover, some examples are given to explain the main results. Finally, we derive several sufficient conditions and nec-essary conditions of approximate solutions which satisfy the property (Pr) for set-valued optimization.