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# Some Theoretical Studies on Vector Optimization

Author: YanAiLing
Tutor: XiuNaiHua
School: Beijing Jiaotong University
Course: Operational Research and Cybernetics
Keywords: Euclidean Jordan algebra vector optimization problem efficient solution weakly efficient solution properly efficient solution spectral function spectral scalar optimization problem robust method ideal Nash-equilibrium point
CLC: O224
Type: PhD thesis
Year: 2008
Quote: 0

### Abstract

 Vector optimization is the optimization problem which solve more than one objectives under some constraints. The theory and methods for the vector optimization are widely used in the areas of modern economic planning, production administration, financial investment, item evaluation, engineering design, transportation, environmental protection, military, etc. In this thesis, we mainly study the theory of vector optimization in three aspects: the spectral scalarization of Euclidean Jordan algebra vector optimization, the robust methods for solving uncertain multiobjective linear optimization problem and the application in game theory.This thesis is divided into four chapters.In Chapter 1, we describe the contents and significance of the vector optimization problem. We also summarize the developments of the vector optimization in three aspects associated with this thesis. Recalling some basic concepts and results on Euclidean Jordan algebra and vector optimization, we outline the contents studied in this thesis.In Chapter 2, we introduce a new scalar function of the vector optimization problem by virtue of the spectral function in Euclidean Jordan algebra. Correspondingly, we define the spectral scalar optimization problem and spectral scalar solution. Based on the speciality of the spectral function, we provide the sufficient conditions such that the spectral function are K—increasing (strictly K—increasing) in Euclidean Jordan algebra, and establish the relationship between the spectral scalar solution set and the K—(weakly) efficient solution set of the Euclidean Jordan algebra vector optimization problem. Particularly, we establish the relationship between the additive spectral scalar solution set and the G properly K—efficient solution (the extended definition of the G properly Pareto efficient solution over the Euclidean Jordan algebra vector optimization problem) set of the Euclidean Jordan algebra vector optimization problem. Meanwhile, the relationship between the trace scalar solution set and the Bo properly K—efficient solution set of the Euclidean Jordan algebra vector optimization problem is es- tablished. At last, we provide the upper semi-continuity and compactness of the spectral scalar solution set mapping under some suitable conditions, and the necessary and sufficient conditions to guarantee the lower semi-continuity of the spectral scalar solution set mapping by virtue of the concept of essential solution.In Chapter 3, we consider the uncertain multiobjective linear optimization problem. Based on robust method, it may be reformulated as the tractable multi-objective optimization problem. At last, two numerical examples are provided to illustrate the efficiency of the robust solutions of the concerned problems.In Chapter 4, by virtue of the solution concepts to three norm scalar optimization problems, we define three new Nash-equilibrium points of the vector game, i.e., 1-normed ideal-Nash equilibrium point, 2-normed idcal-Nash equilibrium point and∞-normed ideal-Nash equilibrium point. And the conditions to guarantee the existence of the three ideal Nash equilibrium points are provided.

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