【作者】 王海军；

【导师】 程曹宗；

【作者基本信息】 北京工业大学 ， 概率论与数理统计， 2012， 博士

【摘要】 对偶理论是最优化问题研究的重要内容，在讨论最优化问题的最优性条件、设计算法等方面有着重要的作用。本文我们主要研究了分式规划问题和向量优化问题的共轭对偶以及极小极大分式规划问题的二阶对偶。同时我们讨论了拟似变分不等式问题的参数适定性。在第2章中，我们讨论一类目标函数是DC函数（凸函数之差）之比，约束条件是有限多个DC函数的不等式约束的分式规划问题。利用Dinkelbach变换将分式规划问题转化为相应的目标函数是DC函数，约束条件是有限多个DC不等式约束的DC规划问题，然后利用共轭对偶的方法构造了该DC规划问题的Fenchel-Lagrange对偶问题，并且在广义内点的约束规则下证明了弱对偶和强对偶定理。应用对偶定理，我们得到了分式规划问题的Farkas型结论，进一步利用相关共轭函数的上境图给出了Farkas型结论的等价表达。在第3章中，我们研究向量优化问题的共轭对偶。一方面，利用凸标量优化问题的共轭对偶方法，构造多目标双层优化问题的两种对偶模型，并且基于原问题的真有效解和弱有效解的概念分别得到了原问题与对偶问题之间的弱对偶和强对偶定理。另一方面，我们利用集值映射的ε-共轭映射的概念，给出了集值优化问题的共轭对偶模型，证明了原问题与对偶问题之间的弱对偶和强对偶定理，并且引入原问题的新的Lagrange函数，得到了ε-鞍点存在的充分必要条件。在第4章中，我们讨论极小极大分式规划问题的二阶对偶。首先引入一类广义二阶凸函数的概念（即二阶(F,α,ρ,d,p)-univex函数）。通过实例说明它是已有文献中的若干二阶凸函数概念的推广。在第2节中，考虑了可微极小极大分式规划问题的两种对偶问题，在相应函数是广义二阶凸的条件下得到了原问题与对偶问题之间的二阶弱对偶、强对偶和严格逆对偶定理。在第3节中，考虑了不可微极小极大分式规划问题的两种对偶问题，得到了相应的二阶对偶定理。在第5章中，我们在Banach空间中讨论拟似变分不等式问题的参数适定性。引入了拟似变分不等式问题的参数适定性和广义参数适定性的概念，首先建立了拟似变分不等式问题的参数适定性的度量刻画，并且在一定条件下证明了拟似变分不等式问题的参数适定性等价于它的解的存在性和唯一性。然后利用逼近解集的非紧测度给出了拟似变分不等式问题的广义参数适定性的充分必要条件。更多还原

【Abstract】 Duality theory is one of the most important contents in optimization prob-lems, which plays an important role in discussing the optimality conditions foroptimization problems, algorithm design and so on. In this thesis, our main aimsare to study the conjugate duality for fractional programming and vector optimiza-tion problems and the second order duality for minimax fractional programming.Besides, we also discuss the parametric well-posedness for quasivariational-likeinequality problems.In Chapter2, we discuss a type of fractional programming problem with aratio of DC functions (diference of convex functions) as objective function andfnitely many inequalities of DC functions as constraints. By using the Dinkelbachtransformation, we convert the fractional programming problem to a DC program-ming problem with the DC objective function and fnitely many DC inequalitiesconstraints. For the DC programming problem, the Fenchel-Lagrange type dualproblem is established, and the weak and strong duality theorems are proved un-der the assumptions of generalized interior point constraint qualifcation. Then wederive the Farkas type results of the fractional programming problem by using theabove duality theorems. Moreover, using the epigraphs of the involved conjugatefunctions, we give an equivalent statement for the Farkas type result.In Chapter3, we study the conjugate duality for vector optimization prob-lems. On the one hand, by using the conjugate duality approach for convex scalaroptimization problems, we construct two dual models for a multiobjective bilevel optimization problem, obtain the weak and strong duality theorems between theprimal problem and its dual problems based on the properly efcient solutionsand weakly efcient solutions of the primal problem, respectively. On the otherhand, we construct a conjugate dual model for a set-valued optimization problemby using the ε-conjugate map of a set-valued function, and prove some weak andstrong duality theorems between the primal problem and its dual problem. Thenwe introduce a new Lagrange function for the primal problem, and obtain somenecessary and sufcient conditions for the existence of the ε-saddle points.In Chapter4, we consider the second order duality for minimax fractionalprogramming problems. First, we introduce a new generalized second order convexfunction, which is called second order (F, α, ρ, d, p)-univex function. Then we givesome examples to illustrate that it is an extension of some known second orderconvex function in the literatures. In section2, we consider two dual problems for adiferentiable minimax fractional programming, and establish second order weak,strong and strictly converse duality theorems between the primal problem andits dual problems under the conditions of above generalized second order convexfunctions. In section3, we consider two dual problems for a nondiferentiableminimax fractional programming, and obtain the related second order dualitytheorems.In Chapter5, we discuss the parametric well-posedness for quasivariational-like inequality problems in Banach spaces. We introduce the notions of the para-metric well-posedness and the parametric well-posedness in generalized sense forquasivariational-like inequality problems. First, we establish some metric char-acterizations of the parametric well-posedness for quasivariational-like inequality problems, and prove that under suitable conditions, the parametric well-posednessis equivalent to the existence and uniqueness of their solutions. Then, we give somenecessary and efcient conditions for the parametric well-posedness in generalizedsense of quasivariational-like inequality problems by using the noncompact mea-sure of the approximating solution sets.更多还原