本文選題：平衡問題 + 向量平衡問題��； 參考：《廣西師范大學(xué)》2017年碩士論文

【摘要】：向量平衡問題是一類普遍的數(shù)學(xué)模型,廣泛應(yīng)用于經(jīng)濟金融、交通運輸、資源分配及工程管理等領(lǐng)域.向量平衡問題解的存在性為向量平衡問題研究的基本問題,具有重要的研究價值.為得到無界集上向量平衡問題解的存在性.一般都需要加上強制性條件,對于變分不等式問題例外簇方法需要條件弱于強制性條件.尚未有文獻利用例外簇方法研究向量平衡問題解的存在性.本文主要研究向量平衡問題,平衡問題,向量優(yōu)化問題解的存在性及解集的有界性.我們主要利用例外簇方法得到相應(yīng)問題存在解的充要條件.本文具體內(nèi)容作如下安排:第一章簡要介紹向量平衡問題及例外簇的歷史背景和發(fā)展概況,并給出文章中涉及的基本定義和引理.第二章主要在自反Banach空間中研究向量平衡問題弱有效解的存在性以及解集的有界性.我們利用對偶向量平衡問題在有界集的限制的解,定義了一個向量平衡問題的例外簇,證明向量平衡問題存在弱有效解等價于不存在例外簇,我們得到向量平衡問題弱有效解存在的充要條件,給出了在自反Banach空間中向量平衡問題弱有效解集有界的充要條件.同時將文獻[17]中平衡問題解集非空的幾個等價條件推廣到向量平衡問題.第三章主要將第二章得到的向量平衡問題解的存在性和解集的非空有界性應(yīng)用到平衡問題,變分不等式問題和凸優(yōu)化問題,列出向量平衡問題退化成平衡問題時,平衡問題例外簇的定義以及平衡問題解集非空有界的等價條件.證明向量平衡問題退化成變分不等式問題時,第二章定義的例外簇與[35],[50]定義的例外簇一致,向量平衡問題退化成凸優(yōu)化問題時,第二章定義的例外簇與[36]定義的例外簇一致.第四章給出向量優(yōu)化問題的例外簇定義,證明若向量優(yōu)化問題無弱有效解:則向量優(yōu)化問題存在例外簇.證明向量優(yōu)化問題弱有效解集非空有界時,則向量優(yōu)化問題不存在例外簇.給出向量優(yōu)化問題存在弱有效解的充要條件和向量優(yōu)化問題弱有效解集非空有界的充要條件.本章推廣了文[47]的部分結(jié)果.我們采用的方法與文[47]不同.
[Abstract]:Vector equilibrium problem is a kind of general mathematical model, which is widely used in the fields of economy and finance, transportation, resource allocation and engineering management. The existence of solution of vector equilibrium problem is the basic problem of vector equilibrium problem, which has important research value. In order to obtain the existence of solutions for vector equilibrium problems on unbounded sets. Generally, it is necessary to add mandatory conditions, and for the exception cluster method of variational inequality, the condition is weaker than the mandatory condition. The existence of solutions for vector equilibrium problems has not been studied by using exceptional cluster method. In this paper, we study the existence of solutions and the boundedness of solutions for vector equilibrium problems, equilibrium problems and vector optimization problems. We mainly use the exceptional cluster method to obtain the necessary and sufficient conditions for the existence of solutions for the corresponding problems. The main contents of this paper are as follows: in Chapter 1, the historical background and development of vector equilibrium problems and exceptional clusters are briefly introduced, and the basic definitions and Lemma involved in this paper are given. In chapter 2, we study the existence of weak efficient solutions and the boundedness of solutions for vector equilibrium problems in reflexive Banach spaces. In this paper, we define an exceptional family of vector equilibrium problems by using the restricted solution of the dual vector equilibrium problem in the bounded set. We prove that the existence of weak efficient solutions for the vector equilibrium problem is equivalent to the absence of the exceptional family. We obtain the necessary and sufficient conditions for the existence of weak efficient solutions for vector equilibrium problems, and give the necessary and sufficient conditions for the bounded set of weak efficient solutions of vector equilibrium problems in reflexive Banach spaces. At the same time, some equivalent conditions of nonempty solutions of equilibrium problems in reference [17] are extended to vector equilibrium problems. In chapter 3, the existence of the solution of the vector equilibrium problem and the nonempty boundedness of the solution set are applied to the equilibrium problem, variational inequality problem and convex optimization problem. The definition of exceptional families for equilibrium problems and the equivalent conditions for the set of solutions of equilibrium problems to be nonempty and bounded. It is proved that when vector equilibrium problem degenerates into variational inequality problem, the exceptional cluster defined in chapter 2 is consistent with that defined in [35] and [50]. When vector equilibrium problem degenerates into convex optimization problem, the exceptional family defined in chapter 2 is consistent with that defined in [36]. In chapter 4, the exceptional cluster definition of vector optimization problem is given, and it is proved that if the vector optimization problem has no weak efficient solution, then the vector optimization problem has an exceptional family. It is proved that when the set of weakly efficient solutions for vector optimization problems is nonempty and bounded, there is no exception family for vector optimization problems. The necessary and sufficient conditions for the existence of weak efficient solutions for vector optimization problems and the necessary and sufficient conditions for nonempty boundedness of the set of weak efficient solutions for vector optimization problems are given. This chapter generalizes some results of [47]. Our approach is different from that in [47].
【學(xué)位授予單位】：廣西師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O224

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本文編號：1860361