集值映射的广义凸性与集值最优化
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摘要
集值优化理论是优化理论和应用的主要研究领域之一。它的理论和方法被广泛应用于微分包含、变分不等式、最优控制、博弈论、经济平衡问题、环境保护、军事决策等领域。对这一问题的研究涉及到集值分析、凸分析、非线性泛函分析、非光滑分析、偏序理论等学科,因此,对它进行研究有重要的理论价值和实际意义。
集值优化问题的最优性条件与解集的结构理论在集值优化理论中占有重要的地位。最优性条件是建立优化算法的重要基础。而凸性和广义凸性在优化理论中起着十分重要的作用。所以,集值映射的广义凸性与集值优化问题的最优性条件的研究成为一个热点问题。
众所周知,集值优化问题的有效解,是关于偏序为非劣意义下的解。因此,这种解的集合一般都相当大,如Geo?rion所见,其中的一些解的性质比较差。所以人们一直致力寻求具有更好形态的有效解–称为真有效解。主要有:超有效解、Benson真有效解、Henig真有效解等。超有效解具有非常好的性质,但是它的存在条件是非常强的。另一方面,为讨论Benson真有效解的标量化、Lagrange乘子、鞍点特征,一般都要求序锥具有紧或弱紧基底。而许多常用的赋范空间中的非负坐标锥都不存在紧或弱紧基底。然而,Henig真有效解既保持了超有效解的主要特征,而存在性条件又比超有效点弱得多;同时它仅要求序锥具有基底。目前,人们对它的研究很少。因此,本论文主要研究集值映射的广义凸性与集值优化问题的Henig真有效性。
全文共分七章,主要内容如下:
在第一章,首先,我们简要介绍了集值优化(向量优化)问题的发展概况和研究意义。其次,阐述了集合的弱有效点、有效点、真有效点之间的关系;介绍了本文研究所需的概念和相关结论。最后,在第1.5节,综述了集值映射的广义凸性和集值优化问题的最优性条件、集值优化问题的近似解与集值优化问题的解集的连通性等方向的研究现状。
在第二章,我们主要研究近似锥次类凸集值映射。首先,我们获得了近似锥次类凸集值映射的一些特征。其次,证明了内部锥类凸集值映射是近似锥次类凸集值映射的特殊情形。最后,我们还得到了近似锥类凸集值映射与内部锥类凸集值映射之间的关系。
在第三章,我们主要研究集值优化问题的Henig真有效性。首先,我们证明了在局部凸空间中,两种Henig真有效点的定义是等价的;详细讨论了Henig真有效点与Benson真有效点之间的关系;给出了Henig真有效点的存在性条件。其次,刻画了近似锥次类凸集值映射优化问题Henig真有效元的标量化、Lagrange乘子、鞍点特征。作为应用,给出了近似锥次类凸集值映射优化问题超有效元的标量化、Lagrange乘子、鞍点特征。
在第四章,我们研究弧式锥凸集值映射优化问题Henig真有效性。首先,在第4.2节给出了相关的概念和引理。然后,在第4.3节证明了弧式锥凸集值映射优化问题Henig真有效元的一个重要特性,即局部Henig真有效元也是Henig真有效元;利用锥方向相依导数,获得了弧式锥凸集值映射优化问题Henig真有效元(强有效元)统一的必要条件和充分条件。最后,在第4.4节讨论了集值映射优化问题Henig真有效解集和超有效解集的连通性问题,在目标函数为弧式锥凸集值映射的条件下,证明了Henig真有效解集和超有效解集是连通的。
在第五章,首先,我们得到了广义锥预不变凸集值映射的一些性质。其次,获得了广义锥预不变凸集值映射优化问题的Henig真有效元、超有效元的充分必要条件;同时建立了广义锥预不变凸集值映射优化问题的真有效性与向量似变分不等式的真有效性之间的密切关系。
在第六章,我们研究集值优化问题的近似解。首先,我们给出了Tammer’s函数的一种有用的表示形式,得到了拟凸集值映射的一个重要性质。其次,在约束集和目标函数不具备任意广义凸性的条件下,获得集值优化问题近似解的标量化特征,并讨论了非凸集值优化问题的近似解与弱有效解之间的关系。最后,我们证明了拟凸集值映射优化问题近似解集是连通的。
在第七章,我们总结了全文的主要结果,并提出了一些有待研究的问题。
Set-valued optimization theory is one of the main research fields of optimizationtheory and applications. Its theories and methods are widely used in the areasof di?erential inclusion, variational inequality, optimal control, game theory, eco-nomic equilibrium problem, environmental protection, military decision making,etc. The study of this topic involves many disciplines, such as: set-valued analysis,convex analysis, nonlinear functional analysis, nonsmooth analysis, partial order-ing theory, and so on. Thus, the research for this topic has important theoreticalvalue and practical significance.
Both optimality condition and the structure theory of solution set of set-valuedoptimization problems are important components in the set-valued optimizationtheory. The optimality condition is an important foundation for developing opti-mization algorithms. And convexity and generalized convexity play a very impor-tant role in the optimization theory. Thus, the research of the generalized convexityof set-valued maps and the optimality conditions of set-valued optimization prob-lems become a hot issue.
It is well known that e?cient solution of set-valued optimization problem is so-lution in the sense of non-inferiority with respect to partial order. Therefore, gener-ally speaking, the set of solutions will be large, just as Geo?rion observed, some ofthe solution set have poor properties. Then, there has been e?orts to seek e?cientsolutions which possess nice properties–called proper e?cient solution. The mainproper e?cient solutions are: super e?cient solution, Benson proper e?cient solu-tion and Henig proper e?cient solution. The super e?cient solution possess niceproperties. But, the existence condition of super e?cient solution is very strong.Meanwhile, for discussing the scalarization theorem and the Lagrangian multipliertheorem in the sense of the Benson proper e?ciency, we have to require that theordering cone with a compact or weak-compact base. But, many positive cones ofnormal space have not a compact or weak-compact base. However, Henig propersolution has many desirable properties, its existence condition is weaker than thatof super e?cient solutions and the ordering cone only requires a base. So far, thereare a few papers which deal with Henig proper solution. Then, the thesis mainlystudy the generalized convexity of set-valued mappings and Henig proper e?ciencyfor set-valued optimization.
The thesis is divided into seven chapters, the main contents are as follows:
In Chapter 1, firstly, we give brief introduction to the development and researchsignificance of set-valued optimization problem. Secondly, the relations amongweak e?cient points, e?cient points and proper e?cient points of a set are given,some concepts and results are presented. Finally, we summarize the developmentof the generalized convexity of set-valued maps and the optimality condition of set-valued optimization problems, the approximate solutions of set-valued optimization problems and the connectedness of the solution set in Section 1.5.
We study the nearly cone-subconvexlike set-valued maps in Chapter 2. Firstly,some characterizations of nearly cone-subconvexlike set-valued maps are derived.Secondly, we prove the ic-cone-convexlike set-valued maps is special case of thenearly cone-subconvexlike set-valued maps. Finally, the relations between nearlycone-convexlikeness and ic-cone-convexlikeness are also given.
In Chapter 3, we mainly study Henig proper e?ciency for set-valued optimiza-tion problems. Firstly, we prove that two definitions of Henig proper e?cient pointare equivalent, discuss the relation between Henig proper e?cient points and Ben-son proper e?cient points, and present the existence conditions of Henig propere?cient point. Under the assumption of nearly cone-subconvexlikeness, a scalar-ization theorem, a Lagrange multiplier theorem and two saddle theorems for Henigproper e?cient pair in set-valued optimization problem are established in Section3.3, Section 3.4 and Section 3.5, respectively. As an interesting application of theresults in this chapter, we establish a scalarization theorem , a Lagrange multipliertheorem and a saddle theorem for super e?cient pair in set-valued optimization.
In Chapter 4, we study Henig proper e?ciency in vector optimization involvingcone-arcwise connected set-valued maps. Firstly, we present some notations andlemmas that are required in the sequel. In Section 4.3, some important propertiesof the cone-arcwise connected set-valued mapping are derived, especially, the globalresult for Henig proper e?cient pair is proven, the unified necessary and su?cientoptimality conditions of Henig proper e?cient pair and strong e?cient pair areobtained by using cone-directed contingent derivative. In Section 4.4, we provethat the set of Henig e?cient solution and the set of super e?cient solution areconnected under objective mappings are cone-arcwise connected set-valued maps.In Chapter 5, firstly, we obtain some properties of the generalized cone-preinvexmapping. Secondly, the necessary and su?cient optimality conditions of Henigproper e?cient pair and super e?cient pair are obtained. Meanwhile, we establishthe closed relationships between proper e?ciency of set-valued optimization andproper e?ciency of vector variational-like inequalities.
In Chapter 6, we study the approximate solutions for vector optimization prob-lem with set-valued maps. Firstly, we present a useful form of Tammer functionand an important characterization of quasiconvex set-valued map in Section 6.2.Secondly, the scalar characterization of approximate solutions is derive withoutimposing any convexity assumption on objective functions in Section 6.3. The re-lations between approximate solutions and weak e?cient solutions are discussed inSection 6.4. Finally, we prove that the set of approximate solutions is connectedunder objective functions are quasiconvex set-valued maps.
In Chapter 7, we summarize the main results of the thesis, and put forwardsome issues to be studied.
集值优化问题的最优性条件与解集的结构理论在集值优化理论中占有重要的地位。最优性条件是建立优化算法的重要基础。而凸性和广义凸性在优化理论中起着十分重要的作用。所以,集值映射的广义凸性与集值优化问题的最优性条件的研究成为一个热点问题。
众所周知,集值优化问题的有效解,是关于偏序为非劣意义下的解。因此,这种解的集合一般都相当大,如Geo?rion所见,其中的一些解的性质比较差。所以人们一直致力寻求具有更好形态的有效解–称为真有效解。主要有:超有效解、Benson真有效解、Henig真有效解等。超有效解具有非常好的性质,但是它的存在条件是非常强的。另一方面,为讨论Benson真有效解的标量化、Lagrange乘子、鞍点特征,一般都要求序锥具有紧或弱紧基底。而许多常用的赋范空间中的非负坐标锥都不存在紧或弱紧基底。然而,Henig真有效解既保持了超有效解的主要特征,而存在性条件又比超有效点弱得多;同时它仅要求序锥具有基底。目前,人们对它的研究很少。因此,本论文主要研究集值映射的广义凸性与集值优化问题的Henig真有效性。
全文共分七章,主要内容如下:
在第一章,首先,我们简要介绍了集值优化(向量优化)问题的发展概况和研究意义。其次,阐述了集合的弱有效点、有效点、真有效点之间的关系;介绍了本文研究所需的概念和相关结论。最后,在第1.5节,综述了集值映射的广义凸性和集值优化问题的最优性条件、集值优化问题的近似解与集值优化问题的解集的连通性等方向的研究现状。
在第二章,我们主要研究近似锥次类凸集值映射。首先,我们获得了近似锥次类凸集值映射的一些特征。其次,证明了内部锥类凸集值映射是近似锥次类凸集值映射的特殊情形。最后,我们还得到了近似锥类凸集值映射与内部锥类凸集值映射之间的关系。
在第三章,我们主要研究集值优化问题的Henig真有效性。首先,我们证明了在局部凸空间中,两种Henig真有效点的定义是等价的;详细讨论了Henig真有效点与Benson真有效点之间的关系;给出了Henig真有效点的存在性条件。其次,刻画了近似锥次类凸集值映射优化问题Henig真有效元的标量化、Lagrange乘子、鞍点特征。作为应用,给出了近似锥次类凸集值映射优化问题超有效元的标量化、Lagrange乘子、鞍点特征。
在第四章,我们研究弧式锥凸集值映射优化问题Henig真有效性。首先,在第4.2节给出了相关的概念和引理。然后,在第4.3节证明了弧式锥凸集值映射优化问题Henig真有效元的一个重要特性,即局部Henig真有效元也是Henig真有效元;利用锥方向相依导数,获得了弧式锥凸集值映射优化问题Henig真有效元(强有效元)统一的必要条件和充分条件。最后,在第4.4节讨论了集值映射优化问题Henig真有效解集和超有效解集的连通性问题,在目标函数为弧式锥凸集值映射的条件下,证明了Henig真有效解集和超有效解集是连通的。
在第五章,首先,我们得到了广义锥预不变凸集值映射的一些性质。其次,获得了广义锥预不变凸集值映射优化问题的Henig真有效元、超有效元的充分必要条件;同时建立了广义锥预不变凸集值映射优化问题的真有效性与向量似变分不等式的真有效性之间的密切关系。
在第六章,我们研究集值优化问题的近似解。首先,我们给出了Tammer’s函数的一种有用的表示形式,得到了拟凸集值映射的一个重要性质。其次,在约束集和目标函数不具备任意广义凸性的条件下,获得集值优化问题近似解的标量化特征,并讨论了非凸集值优化问题的近似解与弱有效解之间的关系。最后,我们证明了拟凸集值映射优化问题近似解集是连通的。
在第七章,我们总结了全文的主要结果,并提出了一些有待研究的问题。
Set-valued optimization theory is one of the main research fields of optimizationtheory and applications. Its theories and methods are widely used in the areasof di?erential inclusion, variational inequality, optimal control, game theory, eco-nomic equilibrium problem, environmental protection, military decision making,etc. The study of this topic involves many disciplines, such as: set-valued analysis,convex analysis, nonlinear functional analysis, nonsmooth analysis, partial order-ing theory, and so on. Thus, the research for this topic has important theoreticalvalue and practical significance.
Both optimality condition and the structure theory of solution set of set-valuedoptimization problems are important components in the set-valued optimizationtheory. The optimality condition is an important foundation for developing opti-mization algorithms. And convexity and generalized convexity play a very impor-tant role in the optimization theory. Thus, the research of the generalized convexityof set-valued maps and the optimality conditions of set-valued optimization prob-lems become a hot issue.
It is well known that e?cient solution of set-valued optimization problem is so-lution in the sense of non-inferiority with respect to partial order. Therefore, gener-ally speaking, the set of solutions will be large, just as Geo?rion observed, some ofthe solution set have poor properties. Then, there has been e?orts to seek e?cientsolutions which possess nice properties–called proper e?cient solution. The mainproper e?cient solutions are: super e?cient solution, Benson proper e?cient solu-tion and Henig proper e?cient solution. The super e?cient solution possess niceproperties. But, the existence condition of super e?cient solution is very strong.Meanwhile, for discussing the scalarization theorem and the Lagrangian multipliertheorem in the sense of the Benson proper e?ciency, we have to require that theordering cone with a compact or weak-compact base. But, many positive cones ofnormal space have not a compact or weak-compact base. However, Henig propersolution has many desirable properties, its existence condition is weaker than thatof super e?cient solutions and the ordering cone only requires a base. So far, thereare a few papers which deal with Henig proper solution. Then, the thesis mainlystudy the generalized convexity of set-valued mappings and Henig proper e?ciencyfor set-valued optimization.
The thesis is divided into seven chapters, the main contents are as follows:
In Chapter 1, firstly, we give brief introduction to the development and researchsignificance of set-valued optimization problem. Secondly, the relations amongweak e?cient points, e?cient points and proper e?cient points of a set are given,some concepts and results are presented. Finally, we summarize the developmentof the generalized convexity of set-valued maps and the optimality condition of set-valued optimization problems, the approximate solutions of set-valued optimization problems and the connectedness of the solution set in Section 1.5.
We study the nearly cone-subconvexlike set-valued maps in Chapter 2. Firstly,some characterizations of nearly cone-subconvexlike set-valued maps are derived.Secondly, we prove the ic-cone-convexlike set-valued maps is special case of thenearly cone-subconvexlike set-valued maps. Finally, the relations between nearlycone-convexlikeness and ic-cone-convexlikeness are also given.
In Chapter 3, we mainly study Henig proper e?ciency for set-valued optimiza-tion problems. Firstly, we prove that two definitions of Henig proper e?cient pointare equivalent, discuss the relation between Henig proper e?cient points and Ben-son proper e?cient points, and present the existence conditions of Henig propere?cient point. Under the assumption of nearly cone-subconvexlikeness, a scalar-ization theorem, a Lagrange multiplier theorem and two saddle theorems for Henigproper e?cient pair in set-valued optimization problem are established in Section3.3, Section 3.4 and Section 3.5, respectively. As an interesting application of theresults in this chapter, we establish a scalarization theorem , a Lagrange multipliertheorem and a saddle theorem for super e?cient pair in set-valued optimization.
In Chapter 4, we study Henig proper e?ciency in vector optimization involvingcone-arcwise connected set-valued maps. Firstly, we present some notations andlemmas that are required in the sequel. In Section 4.3, some important propertiesof the cone-arcwise connected set-valued mapping are derived, especially, the globalresult for Henig proper e?cient pair is proven, the unified necessary and su?cientoptimality conditions of Henig proper e?cient pair and strong e?cient pair areobtained by using cone-directed contingent derivative. In Section 4.4, we provethat the set of Henig e?cient solution and the set of super e?cient solution areconnected under objective mappings are cone-arcwise connected set-valued maps.In Chapter 5, firstly, we obtain some properties of the generalized cone-preinvexmapping. Secondly, the necessary and su?cient optimality conditions of Henigproper e?cient pair and super e?cient pair are obtained. Meanwhile, we establishthe closed relationships between proper e?ciency of set-valued optimization andproper e?ciency of vector variational-like inequalities.
In Chapter 6, we study the approximate solutions for vector optimization prob-lem with set-valued maps. Firstly, we present a useful form of Tammer functionand an important characterization of quasiconvex set-valued map in Section 6.2.Secondly, the scalar characterization of approximate solutions is derive withoutimposing any convexity assumption on objective functions in Section 6.3. The re-lations between approximate solutions and weak e?cient solutions are discussed inSection 6.4. Finally, we prove that the set of approximate solutions is connectedunder objective functions are quasiconvex set-valued maps.
In Chapter 7, we summarize the main results of the thesis, and put forwardsome issues to be studied.
引文
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