集值优化问题的逼近解及二阶最优性条件

项目名称： 集值优化问题的逼近解及二阶最优性条件

项目编号： No.11461044

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 徐义红

作者单位： 南昌大学

项目金额： 36万元

中文摘要： 借助逼近锥族提出一类新的逼近有效点（如逼近超有效点 、逼近Benson真有效点等），从而提出集值优化问题的相应逼近解，如提出逼近超有效解、逼近Benson真有效解的概念. 在近似锥锥次类凸假设下，研究标量化定理，借助逼近Lagrange乘子、逼近鞍点等刻画逼近解，建立逼近解的对偶定理. 针对集值映射利用修改的Dubovitskij-Miljutin切锥引进一种新的二阶切导数,讨论此二阶导数的性质，利用此二阶切导数研究集值优化问题在若干有效元意义下的二阶Kuhn- Tucker最优性条件. 首次引进集值映射在若干有效意义下的二阶次梯度，讨论在某种条件下，二阶次梯度的存在性.讨论二阶次梯度的性质，讨论用二阶次梯度刻画集值优化问题有关解的充要条件.

中文关键词： 逼近解；二阶次梯度；集值映射；严有效性

英文摘要： With the help of approximation families of a cone, a new kind of approximately efficient points (such as approximately superly efficient points、 approximately Benson properly efficient points and so on) will be introduced, thereby approximate solutions for set-valued optimization will be put forward accordingly, such as approximately superly efficient solutions, approximately Benson properly efficient solutions. Under the assumption of near cone-subconvexlikeness for set-valued maps, the scalarization theorems will be investigated. By virtue of approximate Lagrangian multiplier and approximate saddle point, approximate solutions will be characterized,and approximate duality assertions will be established for approximately efficient solutions. A new kind of second-order tangent derivative for a set-valued function will be introduced by means of a modified Dubovitskij-Miljutin cone. Some properties of which will be discussed, Kuhn-Tucker second-order optimality conditions will be obtained for a point pair to be a certain minimizer of set-valued optimization problem. Second-order subgradients for a set-valued map in the sense of some efficiencies (such as strict efficiency and Benson proper efficiency and so on) will be introduced respectively for the first time. Under some conditions their existence theorems will be proved, their properties will be discussed. As applications, the sufficient and necessary optimality conditions will be established for set-valued optimization to obtain relevant efficient solutions.

英文关键词： Approximate solution;second subgradient;set-valued map;strict efficiency