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引进了一种新的二阶组合切锥, 利用它引进了一种新的二阶组合切导数, 称为二阶组合径向切导数, 并讨论了它的性质及它与二阶组合切导数的关系, 借助二阶径向组合切导数, 分别建立了集值优化取得Benson真有效元的最优性充分和必要条件. 相似文献
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在赋范空间中给出了集值映射的二阶切集的概念,利用二阶切集,定义了集值映射的二阶切导数。然后,获得了集值向量优化问题弱极小元的两个二阶最优性必要条件。 相似文献
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Optimality Conditions in
Differentiable Vector Optimization via Second-Order Tangent Sets 总被引:1,自引:0,他引:1
We provide second-order necessary and sufficient conditions for a point to be an efficient element of a set with respect to
a cone in a normed space, so that there is only a small gap between necessary and sufficient conditions. To this aim, we use
the common second-order tangent set and the asymptotic second-order cone utilized by Penot. As an application we establish
second-order necessary conditions for a point to be a solution of a vector optimization problem with an arbitrary feasible
set and a twice Fréchet differentiable objective function between two normed spaces. We also establish second-order sufficient
conditions when the initial space is finite-dimensional so that there is no gap with necessary conditions. Lagrange multiplier
rules are also given.
This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), Project BFM2003-02194.
Online publication 29 January 2004. 相似文献
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In this paper, we develop second-order necessary and sufficient optimality conditions for multiobjective optimization problems with both equality and inequality constraints. First, we generalize the Lin fundamental theorem (Ref. 1) to second-order tangent sets; then, based on the above generalized theorem, we derive second-order necessary and sufficient conditions for efficiency. 相似文献
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In this paper, we study second-order optimality conditions for multiobjective optimization problems. By means of different
second-order tangent sets, various new second-order necessary optimality conditions are obtained in both scalar and vector
optimization. As special cases, we obtain several results found in the literature (see reference list). We present also second-order
sufficient optimality conditions so that there is only a very small gap with the necessary optimality conditions.
The authors thank Professor P.L. Yu and the referees for valuable comments and helpful suggestions. 相似文献
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Akhtar A. Khan 《Optimization》2013,62(6):743-758
In this article we give new second-order optimality conditions in set-valued optimization. We use the second-order asymptotic tangent cones to define second-order asymptotic derivatives and employ them to give the optimality conditions. We extend the well-known Dubovitskii–Milutin approach to set-valued optimization to express the optimality conditions given as an empty intersection of certain cones in the objective space. We also use some duality arguments to give new multiplier rules. By following the more commonly adopted direct approach, we also give optimality conditions in terms of a disjunction of certain cones in the image space. Several particular cases are discussed. 相似文献
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《Optimization》2012,61(5):921-954
ABSTRACTThe paper considers a class of vector optimization problems with cone constrained generalized equations. By virtue of advanced tools of variational analysis and generalized differentiation, a limiting normal cone of the graph of the normal cone constrained by the second-order cone is obtained. Based on the calmness condition, we derive an upper estimate of the coderivative for a composite set-valued mapping. The necessary optimality condition for the problem is established under the linear independent constraint qualification. As a special case, the obtained optimality condition is simplified with the help of strict complementarity relaxation conditions. The numerical results on different examples are given to illustrate the efficiency of the optimality conditions. 相似文献
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Combining results of Avakov about tangent directions to equality constraints given by smooth operators with results of Ben-Tal and Zowe, we formulate a second-order theory for optimality in the sense of Dubovitskii-Milyutin which gives nontrivial conditions also in the case of equality constraints given by nonregular operators. Secondorder feasible and tangent directions are defined to construct conical approximations to inequality and equality constraints which within a single construction lead to first- and second-order conditions of optimality for the problem also in the nonregular case. The definitions of secondorder feasible and tangent directions given in this paper allow for reparametrizations of the approximating curves and give approximating sets which form cones. The main results of the paper are a theorem which states second-order necessary condition of optimality and several corollaries which treat special cases. In particular, the paper generalizes the Avakov result in the smooth case.This research was supported by NSF Grant DMS-91-009324, NSF Grant DMS-91-00043, SIUE Research Scholar Award and Fourth Quarter Fellowship, Summer 1992. 相似文献
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It is well known that second-order cone (SOC) programming can be regarded as a special case of positive semidefinite programming using the arrow matrix. This paper further studies the relationship between SOCs and positive semidefinite matrix cones. In particular, we explore the relationship to expressions regarding distance, projection, tangent cone, normal cone and the KKT system. Understanding these relationships will help us see the connection and difference between the SOC and its PSD reformulation more clearly. 相似文献
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We present a new second-order directional derivative and study its properties. Using this derivative and the parabolic second-order
derivative, we establish second-order necessary and sufficient optimality conditions for a general scalar optimization problem
by means of the asymptotic and parabolic second-order tangent sets to the feasible set. For the sufficient conditions, the
initial space must be finite dimensional. Then, these conditions are applied to a general vector optimization problem obtaining
second-order optimality conditions that generalize the differentiable case. For this aim, we introduce a scalarization, and
the relationships between the different types of solutions to the vector optimization problem and the scalarized problem are
studied.
This research was partially supported by the Ministerio de Educación y Ciencia (Spain), under projects MTM2006-02629 and Ingenio
Mathematica (i-MATH) CSD2006-00032 (Consolider-Ingenio 2010), and by the Consejería de Educación de la Junta de Castilla y
León (Spain), Project VA027B06.
The authors are grateful to the anonymous referees for valuable comments and suggestions. 相似文献
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圆锥规划是一类重要的非对称锥优化问题.基于一个光滑函数,将圆锥规划的最优性条件转化成一个非线性方程组,然后给出求解圆锥规划的光滑牛顿法.该算法只需求解一个线性方程组和进行一次线搜索.运用欧几里得约当代数理论,证明该算法具有全局和局部二阶收敛性.最后数值结果表明算法的有效性. 相似文献
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向量值最优化问题的最优性条件与对偶性 总被引:1,自引:0,他引:1
本文我们首先给出一类向量值优化问题(VP)的正切锥真有效解的定义,在锥方向导数的假设下,讨论了一类单目标问题 的最优性必要条件;然后利用正切锥方向导数定义一类正切锥F-凸函数类,并给出了(VP)正切锥真有效解的充分性条件,最后我们亦讨论了(VP)在正切锥真有效解意义下的对偶性质。 相似文献
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Jinchuan Zhou Jingyong Tang Jein-Shan Chen 《Journal of Optimization Theory and Applications》2017,172(3):802-823
In this paper, we study the parabolic second-order directional derivative in the Hadamard sense of a vector-valued function associated with circular cone. The vector-valued function comes from applying a given real-valued function to the spectral decomposition associated with circular cone. In particular, we present the exact formula of second-order tangent set of circular cone by using the parabolic second-order directional derivative of projection operator. In addition, we also deal with the relationship of second-order differentiability between the vector-valued function and the given real-valued function. The results in this paper build fundamental bricks to the characterizations of second-order necessary and sufficient conditions for circular cone optimization problems. 相似文献
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In this paper, optimality conditions are presented and analyzed for the cardinality-constrained cone programming arising from finance, statistical regression, signal processing, etc. By introducing a restricted form of (strict) Robinson constraint qualification, the first-order optimality conditions for the cardinality-constrained cone programming are established based upon the properties of the normal cone. After characterizing further the second-order tangent set to the cardinality-constrained system, the second-order optimality conditions are also presented under some mild conditions. These proposed optimality conditions, to some extent, enrich the optimization theory for noncontinuous and nonconvex programming problems. 相似文献
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Jean-Paul Penot 《Numerical Functional Analysis & Optimization》2014,35(7-9):1174-1196
Semidefinite positiveness of operators on Euclidean spaces is characterized. Using this characterization, we compute in a direct way the first-order and second-order tangent sets to the cone of semidefinite positive operators on such a space. These characterizations are useful for optimality conditions in semidefinite programming. 相似文献
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借助于Contingent切锥和集值映射的上图而引入的有关集值映射的Contingent切导数,对约束集值优化问题的超有效解建立了最优性Kuhn Tucker必要及充分性条件,借此建立了向量集值优化超有效解的Wolfe型和Mond Weir型对偶定理. 相似文献