约束优化问题的若干对偶以及微分性研究
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摘要
本文研究了广义平衡问题、分式优化问题、复合凸优化问题的对偶理论以及Farkas-型结果,集值优化问题的高阶最优性条件和高阶Fritz John型最优性条件，参数向量优化问题扰动映射的广义可微性。本文分为七章，具体如下
第一章，首先回顾了广义平衡问题与若干优化问题对偶理论的研究近况，随后阐明了向量优化问题的最优性与灵敏性的研究现状，最后介绍了本文的研究动机以及主要工作。
第二章，考虑带有DC函数的广义平衡问题。借助Fenchel共轭函数的方法，首先引入了广义平衡问题的对偶问题，并得到了相应的弱对偶与强对偶。随后借助得到的对偶理论刻画了广义平衡问题的Farkas-型结果。最后运用研究广义平衡问题的方法，研究了凸优化问题和广义变分不等式问题。
第三章，考虑约束分式优化问题。借助Dinkelbach[1]的方法，首先将分式优化问题转化为约束优化问题。同时针对不同的情形，引入了约束优化问题的两种不同的对偶问题。随后借助共轭函数上图的性质，研究了约束优化问题的对偶理论。最后借助约束优化问题的对偶理论研究了分式优化问题的Farkas-型结果。
第四章，考虑复合凸优化问题。在局部凸空间中，借助共轭函数上图的性质，引入了一些约束品性并讨论它们的一些性质。随后引入了复合凸优化问题的对偶问题。最后借助引入的约束品性刻画了复合凸优化问题的强对偶、稳定强对偶以及全对偶。同时说明了所得结果推广和改进了相关文献的结果。
第五章，首先引入了集值映射的Studniarski导数并研究了相应的性质。随后利用集值映射的Studniarski导数研究了约束集值优化问题严格局部有效解的高阶最优性条件以及高阶Fritz John型最优性条件。进一步的，研究了集值映射的Studniarski导数与其剖面映射的Studniarski导数之间的关系。同时也得到了关于参数向量优化问题的Studniarski导数的一些灵敏性结果。
第六章，考虑参数集值优化问题。借助二阶方向紧性的假设，首先研究了两个集值映射复合的二阶相依导数以及两个集值映射和的二阶相依导数的计算法则。随后利用得到的计算法则研究了一类参数集值优化问题的二阶灵敏性结果并得到了参数集值优化问题的扰动映射与弱扰动映射的二阶相依导数的一个具体表达式。最后研究了参数向量优化问题的最优解映射以及最优值映射的广义二阶相依上图导数的计算法则。
第七章，首先对本文的研究内容进行了一个简单的总结，随后提出了一些本文研究中存在的遗留问题以及今后打算思考和研究的问题。
In this thesis, the duality assertions and Farkas-type results for generalizedequilibrium problems, fractional programming problems and composed convexoptimization problems are investigated. And the higher-order optimality conditions andhigher-order Fritz John type optimality conditions for a set-valued optimization problemare also investigated. Moreover, the generalized differentiability of perturbationmappings for parametric vector optimization problems are also investigated. This thesisis divided into seven chapters. It is organized as follows:
In Chapter1, at first, the development and researches on the topic of duality forgeneralized equilibrium problems and some optimization problems are recalled. Then,the development and researches on the topic of optimality and sensitivity for vectoroptimization problems are reviewed. Finally, we give the motivations and list the mainresearch works.
In Chapter2, a generalized equilibrium problem with DC functions is considered.By using the method of Fenchel conjugate function, a dual scheme for the generalizedequilibrium problem is introduced. And the weak and strong duality assertions areobtained. Then, by using the obtained duality assertions, some Farkas-type resultswhich characterize the optimal value of the generalized equilibrium problem are given.Finally, the proposed approach is applied to a convex optimization problem and ageneralized variational inequality problem.
In Chapter3, a constrained fractional programming problem is considered. Byusing an idea due to Dinkelbach[1], we first associate the fractional programmingproblem with a constrained optimization problem. Adopting different tactics, two typesof dual problems of the constrained optimization problem are constructed. After that, byusing the properties of the epigraph of the conjugate functions, the duality assertions forthe constrained optimization problem are investigated. Finally, by using the dualityassertions, some Farkas-type results for the fractional programming problem areobtained.
In Chapter4, a composed convex optimization problem is considered. By using theproperties of the epigraph of the conjugated functions, some new constraintqualifications are introduced. Then, by using these new constraint qualifications, somenecessary and sufficient conditions which characterize the stable strong and total dualities for the composed convex optimization problem are obtained. Moreover, weshow that our general results encompass as special cases some recently obtained resultsin the literature.
In Chapter5, Studniarski derivatives of set-valued maps are first defined and theirproperties are discussed. Then, by virtue of these derivatives and strict local minimality,higher-order optimality conditions and higher-order Fritz John type optimalityconditions are obtained for a set-valued optimization problem whose constraintcondition is determined by a set-valued map. Moreover, the relationships betweenStudniarski derivative of a set-valued map and its profile map are investigated. Somesensitivity results for a parametrized vector optimization are also obtained.
In Chapter6, a family of parameterized set-valued optimization problems, whoseconstraint set depends on a parameter, are considered. Some calculus rules are obtainedfor calculating the second-order contingent derivatives of the composition and sum oftwo set-valued mappings. Then, by using these calculus rules, some results concerningsensitivity analysis are established, and an explicit expression for the second-ordercontingent derivative of the (weak) perturbation mapping in the set-valued optimizationproblems is obtained. Moreover, some calculus rules of the generalized second-ordercontingent epiderivative for frontier and efficient solution maps in parametric vectoroptimization problems are established.
In Chapter7, the results of this thesis are briefly summarized. Some problems whichare remained and thought over in future are put forward.
第一章，首先回顾了广义平衡问题与若干优化问题对偶理论的研究近况，随后阐明了向量优化问题的最优性与灵敏性的研究现状，最后介绍了本文的研究动机以及主要工作。
第二章，考虑带有DC函数的广义平衡问题。借助Fenchel共轭函数的方法，首先引入了广义平衡问题的对偶问题，并得到了相应的弱对偶与强对偶。随后借助得到的对偶理论刻画了广义平衡问题的Farkas-型结果。最后运用研究广义平衡问题的方法，研究了凸优化问题和广义变分不等式问题。
第三章，考虑约束分式优化问题。借助Dinkelbach[1]的方法，首先将分式优化问题转化为约束优化问题。同时针对不同的情形，引入了约束优化问题的两种不同的对偶问题。随后借助共轭函数上图的性质，研究了约束优化问题的对偶理论。最后借助约束优化问题的对偶理论研究了分式优化问题的Farkas-型结果。
第四章，考虑复合凸优化问题。在局部凸空间中，借助共轭函数上图的性质，引入了一些约束品性并讨论它们的一些性质。随后引入了复合凸优化问题的对偶问题。最后借助引入的约束品性刻画了复合凸优化问题的强对偶、稳定强对偶以及全对偶。同时说明了所得结果推广和改进了相关文献的结果。
第五章，首先引入了集值映射的Studniarski导数并研究了相应的性质。随后利用集值映射的Studniarski导数研究了约束集值优化问题严格局部有效解的高阶最优性条件以及高阶Fritz John型最优性条件。进一步的，研究了集值映射的Studniarski导数与其剖面映射的Studniarski导数之间的关系。同时也得到了关于参数向量优化问题的Studniarski导数的一些灵敏性结果。
第六章，考虑参数集值优化问题。借助二阶方向紧性的假设，首先研究了两个集值映射复合的二阶相依导数以及两个集值映射和的二阶相依导数的计算法则。随后利用得到的计算法则研究了一类参数集值优化问题的二阶灵敏性结果并得到了参数集值优化问题的扰动映射与弱扰动映射的二阶相依导数的一个具体表达式。最后研究了参数向量优化问题的最优解映射以及最优值映射的广义二阶相依上图导数的计算法则。
第七章，首先对本文的研究内容进行了一个简单的总结，随后提出了一些本文研究中存在的遗留问题以及今后打算思考和研究的问题。
In this thesis, the duality assertions and Farkas-type results for generalizedequilibrium problems, fractional programming problems and composed convexoptimization problems are investigated. And the higher-order optimality conditions andhigher-order Fritz John type optimality conditions for a set-valued optimization problemare also investigated. Moreover, the generalized differentiability of perturbationmappings for parametric vector optimization problems are also investigated. This thesisis divided into seven chapters. It is organized as follows:
In Chapter1, at first, the development and researches on the topic of duality forgeneralized equilibrium problems and some optimization problems are recalled. Then,the development and researches on the topic of optimality and sensitivity for vectoroptimization problems are reviewed. Finally, we give the motivations and list the mainresearch works.
In Chapter2, a generalized equilibrium problem with DC functions is considered.By using the method of Fenchel conjugate function, a dual scheme for the generalizedequilibrium problem is introduced. And the weak and strong duality assertions areobtained. Then, by using the obtained duality assertions, some Farkas-type resultswhich characterize the optimal value of the generalized equilibrium problem are given.Finally, the proposed approach is applied to a convex optimization problem and ageneralized variational inequality problem.
In Chapter3, a constrained fractional programming problem is considered. Byusing an idea due to Dinkelbach[1], we first associate the fractional programmingproblem with a constrained optimization problem. Adopting different tactics, two typesof dual problems of the constrained optimization problem are constructed. After that, byusing the properties of the epigraph of the conjugate functions, the duality assertions forthe constrained optimization problem are investigated. Finally, by using the dualityassertions, some Farkas-type results for the fractional programming problem areobtained.
In Chapter4, a composed convex optimization problem is considered. By using theproperties of the epigraph of the conjugated functions, some new constraintqualifications are introduced. Then, by using these new constraint qualifications, somenecessary and sufficient conditions which characterize the stable strong and total dualities for the composed convex optimization problem are obtained. Moreover, weshow that our general results encompass as special cases some recently obtained resultsin the literature.
In Chapter5, Studniarski derivatives of set-valued maps are first defined and theirproperties are discussed. Then, by virtue of these derivatives and strict local minimality,higher-order optimality conditions and higher-order Fritz John type optimalityconditions are obtained for a set-valued optimization problem whose constraintcondition is determined by a set-valued map. Moreover, the relationships betweenStudniarski derivative of a set-valued map and its profile map are investigated. Somesensitivity results for a parametrized vector optimization are also obtained.
In Chapter6, a family of parameterized set-valued optimization problems, whoseconstraint set depends on a parameter, are considered. Some calculus rules are obtainedfor calculating the second-order contingent derivatives of the composition and sum oftwo set-valued mappings. Then, by using these calculus rules, some results concerningsensitivity analysis are established, and an explicit expression for the second-ordercontingent derivative of the (weak) perturbation mapping in the set-valued optimizationproblems is obtained. Moreover, some calculus rules of the generalized second-ordercontingent epiderivative for frontier and efficient solution maps in parametric vectoroptimization problems are established.
In Chapter7, the results of this thesis are briefly summarized. Some problems whichare remained and thought over in future are put forward.
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