集值优化问题的逼近解及二阶最优性条件

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.

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.

集值优化理论在微分包含、逼近论、变分等领域均有广泛的应用，而集值优化问题的最优性条件是其中的重要组成部分，是建立现代优化算法的重要基础..借助逼近锥族提出了一类新的逼近超有效点，从而提出了集值优化问题的逼近超有效解. 在近似锥-次类凸假设下，得到了研究标量化定理，借助逼近Lagrange乘子建立了逼近超有效解的最优性条件. .针对集值映射利用修改的Dubovitskij-Miljutin切锥分别引进了二阶切上图导数和二阶M-切导数, 分别讨论了这两类二阶切导数的性质，分别利用这两类二阶切导数建立了集值优化问题在弱有效元、Benson真有效元意义下的二阶Kuhn- Tucker最优性条件. 在最优性条件的表达式中目标函数和约束函数是分开的，与经典的非导数型最优性条件相吻合，这样的优点是目标函数的二阶切导数可通过约束函数的二阶切导数来表示..首次引进了集值映射在弱有效意义下的二阶次梯度，得到了二阶次梯度的存在性定理. 得到了二阶次梯度的性质，建立了用二阶次梯度刻画集值优化问题弱有效解的充要条件..利用高阶相依切导数定量分析了扰动映射在Henig意义下的性质. 当序锥具有紧基时，利用凸集分离定理得到了高阶灵敏度分析结果..引进了锥-次弧连通集值映射的概念，分别举例说明了锥-次弧连通集值映射是锥-弧连通集值映射的真推广和弧连通集是凸集的真推广. 借助广义二阶切上图导数给出了集值优化问题取得全局真有效元的必要条件. 当目标函数是锥次弧连通的时，得到了集值优化问题取得全局真有效元的充分条件..在Hausdorff局部凸拓扑线性空间中考虑了二元集值函数近似严有效鞍点问题, 在近似锥-次类凸(凹)假设下, 利用凸集分离定理得到二元集值函数取得近似严有效元的松弛型鞍点的必要条件, 利用标量化定理得到充分条件. .我们所的结果对集值优化问题理论的发展和完善有着积极的意义.