集值优化最优性条件与稳定性问题的研究
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摘要
集值优化理论在不动点、变分学、微分包含、最优控制、数理经济学等领域有着广泛的应用,是目前应用数学领域中备受关注的热点之一。对这一问题的研究涉及到集值分析、凸分析、线性与非线性分析、非光滑分析、拓扑向量格、偏序理论等数学分支,有重要的学术价值和相当的难度。
集值优化问题的最优性条件和稳定性在集值优化理论中占有重要的地位。最优性条件是建立现代优化算法的重要基础;稳定性是优化理论的重要组成部分,向量优化的稳定性通过研究各种适定性取得了丰富的结果。但是,关于研究集值优化问题的稳定性的文献很少见到(Huang X.X.仅研究无约束参数集值优化问题在上半连续意义下的稳定性)。本文主要对集值优化问题的各种有效性的最优性条件及集值优化问题的有效解集和有效点集的稳定性进行了较为深入的研究。文章通过集值映射的导数、广义梯度及集值优化问题的鞍点刻画集值优化问题的最优性条件;并且集中研究集值优化问题的有效解集在各种上半连续意义下的稳定性及有效点集在次微分意义下的稳定性。具体内容如下:
● 一方面,在赋范空间中,讨论集值优化问题的有效元的导数型最优性条件。给出了可微Γ-拟凸集值映射的概念。当目标映射和约束映射的下方向导数存在时,在近似锥次类凸假设下利用有效点的性质和凸集分离定理得到了集值优化问题有效元的导数型Kuhn-Tucker必要条件;在可微Γ-拟凸性的假设下得到Kuhn-Tucker最优性充分条件;此外利用集值映射沿弱方向锥的导数特性给出有效解最优性的另一种刻画。另一方面,在局部凸拓扑向量空间,利用Dinh
T.L.给出的集值映射的下半可微性定义了集值映射的导数。在凸性及拟凸性的假设下,利用凸集分离定理得到了集值优化问题的超有效元导数型Kuhn-Tucker最优性充分和必要条件。
● 在局部凸拓扑向量空间中,利用强鞍点和严鞍点刻画了集值优化问题的强有效元与严有效元的最优性条件。首次定义了集值优化问题的强鞍点和严鞍点,给出了强鞍点和严鞍点的等价刻画;在一定凸性条件下,通过凸集分离定理及强鞍点、严鞍点的性质分别得到了强鞍点和严鞍点的最优性条件;考虑了Lagrange型对偶问题,分别给出了强有效、严有效意义下的弱对偶、逆对偶、强对偶定理,并得到了分别由强鞍点、严鞍点刻画的强有效元、严有效元的最优性条件。
● 在锥偏序的Banach空间中,讨论由广义梯度刻画的集值优化问题严有
The theory of set-valued optimization, research on which involve in such mathematical branches as set-valued analysis, convex analysis, linear and nonlinear analysis, nonsmooth analysis, topolocial vector lattice, partial ordering theory, is one of focal point problems in the vector optimization field and finds wide applications in fixed point theory, variation problems, differential inclusions, control theory and mathematical economics. Therefore, the research for them has important learning value and certain degree of difficulty.Both the optimality conditions and stability of set-valued optimization problems are important components in the theory of set-valued optimization. The optimality conditions of set-valued optimization problems is an important foundation for developing modern algorithms;The stability of optimization problems is necessary subjects in the theory of optimization, The stability of vector optimization problems has made the rich results through studying the well-posedness of problems. But the stability of set-valued optimization problems has not been established (Huang X.X. only studies the stability of unconstraint set-valued optimization problems in the sense of upper-semicontinuty). This paper is devoted to character the optimality conditions of set-valued optimization problems in the sense of various efficiency and study systematically the stability of the sets of efficient points and efficient solutions of set-valued optimization problems. The research is carried on from two aspects. One is, based on establishing the derivative and generalized gradient of set-valued maps and saddle points of set-valued optimization problems, to character the optimality of set-valued optimization problems;the other is to study the stability of efficient solutions of set-valued optimization problems in sense of various semicontinuity and the stability of efficient points of problems in sense of subdifferential. The main points of this paper is as follows:
? The optimality conditions of set-valued optimization problems with derivatives are established under efficiency in normed linear space. The concept of F-quasi-convexity is introduced. When the lower direct derivatives of objectives maps and constrained maps exist, under the assumption of nearly cone-subconvexlikeness, by using properties of set of efficient points and a separation theorem for convex sets, Kuhn-Tucker necessary conditions are obtained for set-valued optimization problems in sense of efficiency. Under the assumption of F-quasi-convexity, Kuhn-Tucker sufficient condition is obtained for set-valued optimization problems in sense of efficiency;moreover, another characterization of optimality condition for efficiency is presented by using the properties of lower direct derivative of set-valued maps at weak feasible directs. On the other hand, the derivative of set-valued is given in locally convex linear space by applying the lower semidifferentiable for set-valued maps denned by Dinh T.L. in locally convex linear space. Under convexity and quasi-convexity assumption, by applying the separation theorem, Kuhn-Tucker necessary and sufficient conditions are presented for super efficiency.? The optimality conditions of strict efficiency and strong efficiency for set-valued optimization problems are presented by the strict saddle points and strong saddle points, respectively, the strict saddle points and strong saddle points of set-valued optimization problems are defined and the equivalent characterization of strict saddle points and strong saddle points are given, respectively;The optimality conditions of strict saddle points and strong saddle points are obtained by applying the separation theorem and properties of strict and strong saddle points, respectively;The Lagrange type duality problems for efficient element of set-valued optimization under strict efficiency and strong efficiency are investigated and the weak duality , strong duality, and inverse duality theorem are presented under strict efficiency and strong efficiency, then the optimality conditions of strict efficiency and strong efficiency are presented.
? The optimality conditions of strict efficiency and super efficiency for set-valued optimization problems are presented by generalized gradient under strict efficiency and super efficiency, respectively. The generalized gradient of set-valued maps under strict efficiency and strong efficiency is introduced by applying contingent epiderivative of set-valued maps. Under convexity assumption, the existences of generalized gradient of set-valued maps under strict efficiency and super efficiency are proved and the optimality conditions of strict efficiency and super efficiency is obtained, respectively.? The Well-posedness and stability of set-valued optimization problems are investigated in sense of upper semicontinuious. The pointwise well-posedness of set-valued optimization problems is presented in metric vector space, an equivalent characterization is given, The pointwise well-posedness of one type set-valued optimization problems is verified, and Ekeland's vari-ational principle for set-valued maps is proved;The B-well-posedness of set-valued optimization problems is also presented, the B-well-posedness is characterized by applying asymptotically minimizing sequence, the fact that the projects of any asymptotically minimizing sequence of set-valued optimization problems being B-well-posedness on domain space converge to the set of efficient solutions of this problem is shown, and stability of parametric set-valued optimization problems is obtained under Hausdoff semicontinu-ous.? The stability of set-valued optimization problems in sense of semi-continuity and subdifferential are investigated in normed space, the sub-differential of set-valued maps is defined in sense of super efficiency and strict efficiency, under suitable conditions, the existence and properties of subdifferential are proved by applying the cone separation theorem;When domination cone or domination cone, constrainted set, and objective maps is perturbed, the stability of set-valued optimization problems is investigated in the sense of subdifferential defined under super efficiency and in sense of
semicontinuity;the stability of set of strictly efficient points of set-valued optimization problems is investigated in the sense of subdifferential defined under strict efficiency.
集值优化问题的最优性条件和稳定性在集值优化理论中占有重要的地位。最优性条件是建立现代优化算法的重要基础;稳定性是优化理论的重要组成部分,向量优化的稳定性通过研究各种适定性取得了丰富的结果。但是,关于研究集值优化问题的稳定性的文献很少见到(Huang X.X.仅研究无约束参数集值优化问题在上半连续意义下的稳定性)。本文主要对集值优化问题的各种有效性的最优性条件及集值优化问题的有效解集和有效点集的稳定性进行了较为深入的研究。文章通过集值映射的导数、广义梯度及集值优化问题的鞍点刻画集值优化问题的最优性条件;并且集中研究集值优化问题的有效解集在各种上半连续意义下的稳定性及有效点集在次微分意义下的稳定性。具体内容如下:
● 一方面,在赋范空间中,讨论集值优化问题的有效元的导数型最优性条件。给出了可微Γ-拟凸集值映射的概念。当目标映射和约束映射的下方向导数存在时,在近似锥次类凸假设下利用有效点的性质和凸集分离定理得到了集值优化问题有效元的导数型Kuhn-Tucker必要条件;在可微Γ-拟凸性的假设下得到Kuhn-Tucker最优性充分条件;此外利用集值映射沿弱方向锥的导数特性给出有效解最优性的另一种刻画。另一方面,在局部凸拓扑向量空间,利用Dinh
T.L.给出的集值映射的下半可微性定义了集值映射的导数。在凸性及拟凸性的假设下,利用凸集分离定理得到了集值优化问题的超有效元导数型Kuhn-Tucker最优性充分和必要条件。
● 在局部凸拓扑向量空间中,利用强鞍点和严鞍点刻画了集值优化问题的强有效元与严有效元的最优性条件。首次定义了集值优化问题的强鞍点和严鞍点,给出了强鞍点和严鞍点的等价刻画;在一定凸性条件下,通过凸集分离定理及强鞍点、严鞍点的性质分别得到了强鞍点和严鞍点的最优性条件;考虑了Lagrange型对偶问题,分别给出了强有效、严有效意义下的弱对偶、逆对偶、强对偶定理,并得到了分别由强鞍点、严鞍点刻画的强有效元、严有效元的最优性条件。
● 在锥偏序的Banach空间中,讨论由广义梯度刻画的集值优化问题严有
The theory of set-valued optimization, research on which involve in such mathematical branches as set-valued analysis, convex analysis, linear and nonlinear analysis, nonsmooth analysis, topolocial vector lattice, partial ordering theory, is one of focal point problems in the vector optimization field and finds wide applications in fixed point theory, variation problems, differential inclusions, control theory and mathematical economics. Therefore, the research for them has important learning value and certain degree of difficulty.Both the optimality conditions and stability of set-valued optimization problems are important components in the theory of set-valued optimization. The optimality conditions of set-valued optimization problems is an important foundation for developing modern algorithms;The stability of optimization problems is necessary subjects in the theory of optimization, The stability of vector optimization problems has made the rich results through studying the well-posedness of problems. But the stability of set-valued optimization problems has not been established (Huang X.X. only studies the stability of unconstraint set-valued optimization problems in the sense of upper-semicontinuty). This paper is devoted to character the optimality conditions of set-valued optimization problems in the sense of various efficiency and study systematically the stability of the sets of efficient points and efficient solutions of set-valued optimization problems. The research is carried on from two aspects. One is, based on establishing the derivative and generalized gradient of set-valued maps and saddle points of set-valued optimization problems, to character the optimality of set-valued optimization problems;the other is to study the stability of efficient solutions of set-valued optimization problems in sense of various semicontinuity and the stability of efficient points of problems in sense of subdifferential. The main points of this paper is as follows:
? The optimality conditions of set-valued optimization problems with derivatives are established under efficiency in normed linear space. The concept of F-quasi-convexity is introduced. When the lower direct derivatives of objectives maps and constrained maps exist, under the assumption of nearly cone-subconvexlikeness, by using properties of set of efficient points and a separation theorem for convex sets, Kuhn-Tucker necessary conditions are obtained for set-valued optimization problems in sense of efficiency. Under the assumption of F-quasi-convexity, Kuhn-Tucker sufficient condition is obtained for set-valued optimization problems in sense of efficiency;moreover, another characterization of optimality condition for efficiency is presented by using the properties of lower direct derivative of set-valued maps at weak feasible directs. On the other hand, the derivative of set-valued is given in locally convex linear space by applying the lower semidifferentiable for set-valued maps denned by Dinh T.L. in locally convex linear space. Under convexity and quasi-convexity assumption, by applying the separation theorem, Kuhn-Tucker necessary and sufficient conditions are presented for super efficiency.? The optimality conditions of strict efficiency and strong efficiency for set-valued optimization problems are presented by the strict saddle points and strong saddle points, respectively, the strict saddle points and strong saddle points of set-valued optimization problems are defined and the equivalent characterization of strict saddle points and strong saddle points are given, respectively;The optimality conditions of strict saddle points and strong saddle points are obtained by applying the separation theorem and properties of strict and strong saddle points, respectively;The Lagrange type duality problems for efficient element of set-valued optimization under strict efficiency and strong efficiency are investigated and the weak duality , strong duality, and inverse duality theorem are presented under strict efficiency and strong efficiency, then the optimality conditions of strict efficiency and strong efficiency are presented.
? The optimality conditions of strict efficiency and super efficiency for set-valued optimization problems are presented by generalized gradient under strict efficiency and super efficiency, respectively. The generalized gradient of set-valued maps under strict efficiency and strong efficiency is introduced by applying contingent epiderivative of set-valued maps. Under convexity assumption, the existences of generalized gradient of set-valued maps under strict efficiency and super efficiency are proved and the optimality conditions of strict efficiency and super efficiency is obtained, respectively.? The Well-posedness and stability of set-valued optimization problems are investigated in sense of upper semicontinuious. The pointwise well-posedness of set-valued optimization problems is presented in metric vector space, an equivalent characterization is given, The pointwise well-posedness of one type set-valued optimization problems is verified, and Ekeland's vari-ational principle for set-valued maps is proved;The B-well-posedness of set-valued optimization problems is also presented, the B-well-posedness is characterized by applying asymptotically minimizing sequence, the fact that the projects of any asymptotically minimizing sequence of set-valued optimization problems being B-well-posedness on domain space converge to the set of efficient solutions of this problem is shown, and stability of parametric set-valued optimization problems is obtained under Hausdoff semicontinu-ous.? The stability of set-valued optimization problems in sense of semi-continuity and subdifferential are investigated in normed space, the sub-differential of set-valued maps is defined in sense of super efficiency and strict efficiency, under suitable conditions, the existence and properties of subdifferential are proved by applying the cone separation theorem;When domination cone or domination cone, constrainted set, and objective maps is perturbed, the stability of set-valued optimization problems is investigated in the sense of subdifferential defined under super efficiency and in sense of
semicontinuity;the stability of set of strictly efficient points of set-valued optimization problems is investigated in the sense of subdifferential defined under strict efficiency.
引文
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