Some Problems On Multiobjective Optimization

In the studying of the problem of the multiobjective optimization (also being called vector optimization), they are very important issues to study the stability of the efficient solutions and the weak efficient solutions of the problem of the multiobjective optimization, and to study that the efficient solutions of the problem of the multiobjective optimization are the approximation of the solutions of the corresponding scalar optimization problem. This thesis achieves the following results in the study of these two issues of the theory of the multiobjective optimization: (1)investigate the sensitivity analysis of the multiobjective optimization under the condition that the ordering cone has a weak compact basis; (2)give out the stability analysis of the efficient solutions and the weak efficient solutions under the condition that the ordering cone is normal but has not the bounded basis; (3) introduce the new concept of the semi-convergence of the sequence of sets and investigate the property of the semi-convergence of the sequence of sets, and study the compactness and connection of the limited set; (4)use the solution of the scalar optimization problem to approach the efficient solution of the problem of the multiobjective optimization in infinite dimensional space..This thesis is divided into six chapters. Chapter 1 illustrate the concept of the multiobjective optimization and the motive of the project for research work.In chapter 2, we deal with the relationship between the contingent derivatives of set-valued map F and the contingent derivatives of F+P under the condition that the ordering cone has a weakly compact base, and we introduce a new concept of upper locally Lipschitz of the set-valued map, by this concept, we obtain a new result of the sensitivity analysis in multiobjective optimization in finite dimensional space.Chapter 3 investigates the stability of the sets of efficient points in the sense of the convergence of Painlevé-Kuratowski [12] in the problem of multiobjective optimization under the condition that the ordering cone is regular but has not a bounded base, improving the main result of [12]. This chapter also investigates the stability of the sets of the efficient points and the weak efficient points of problems in multiobjective optimization of vector-valued map and set-valued map.In chapter 4 a new style of dilating ordering cone is introduced, and we investigate the character of this dilating ordering cone. With the help with the new concept, this paper also gets the results that using the solution of the corresponding scalar optimization problem to approach the efficient solution of the problem of the multiobjective optimization in infinite space.In chapter 5, with the help of another style of dilating ordering cone which introduced by Henig[28] and by using the thought of Helbig[27], we also deals with the results that using the solution of the corresponding scalar optimization problem to approach the efficient solution of the problem of the multiobjective optimization in infinite space. This is the extension of the famous theory of Arrow, Barankin, Blakewell[31] when the set is not convex.In chapter 6 we introduce the new concept of upper semi-convergence, lower semi -convergence, convergence of sequences of sets in topological space. We give out the properties of semi-convergent sequences of sets, and discuss the connectedness and compactness of limit set, and also discuss the relationship among the semi-convergence, Painlevé-Kuratowski convergence, and bounded Hausdorff convergence. This chapter reveals that we provide a new tool for stability analysis.Chapter 7 summarize the results of all this thesis and look forward the deeper research about this kind of work.