广义高阶锥方向导数及对集值优化的应用

数学物理学报 ›› 2011, Vol. 31 ›› Issue (4): 902-909.

广义高阶锥方向导数及对集值优化的应用

王其林^1,2|李声杰¹

1.重庆大学数学与统计学院重庆 400044; |2.重庆交通大学理学院重庆 400074

收稿日期:2009-03-10 修回日期:2010-10-15 出版日期:2011-08-25 发布日期:2011-08-25
基金资助:
国家自然科学基金( 10871216, 11071267)和重庆市教委科学技术研究项目(KJ100419)资助

Generalized |Higher-order Cone-directed Derivatives and Applications to Set-valued Optimization

WANG Qi-Lin^1,2, LI Shng-Jie¹

1.College of Mathematics and Statistics, Chongqing University, Chongqing |400044|
2.College of Sciences, |Chongqing Jiaotong University, Chongqing 400074

Received:2009-03-10 Revised:2010-10-15 Online:2011-08-25 Published:2011-08-25
Supported by:
国家自然科学基金( 10871216, 11071267)和重庆市教委科学技术研究项目(KJ100419)资助

摘要/Abstract

摘要：

该文引入了一个集合的广义高阶相依(邻接)集和集值映射的广义高阶锥方向相依(邻接)导数, 基于这些导数概念, 获得了约束条件分别由一个集合和集值映射决定的集值优化问题的广义高阶必要和充分最优性条件.

关键词: 广义高阶相依(邻接)集, 广义高阶锥方向相依(邻接)导数, 集值优化问题, 弱极小元, 广义高阶最优性条件

Abstract:

In this paper, a generalized higher-order contingent (adjacent) set for a set and a generalized higher-order cone-directed contingent (adjacent) derivative for a set-valued map are introduced. By virtue of the derivatives, generalized higher-order necessary and sufficient conditions are obtained for a set-valued optimization problem, whose constraint condition is determined by a fixed set or a set-valued map, respectively.

Key words: Generalized higher-order contingent (adjacent) sets, Generalized higher-order cone-directed contingent (adjacent) derivatives,
Set-valued optimization, Weak minimizer, Generalized higher-order optimality conditions

中图分类号:

90C26

王其林, 李声杰. 广义高阶锥方向导数及对集值优化的应用[J]. 数学物理学报, 2011, 31(4): 902-909.

WANG Qi-Lin, LI Shng-Jie. Generalized |Higher-order Cone-directed Derivatives and Applications to Set-valued Optimization[J]. Acta mathematica scientia,Series A, 2011, 31(4): 902-909.

参考文献

[1] Aubin J P, Frankowska H. Set-Valued Analysis. Boston: Biekhäuser, 1990

[2] Certh C, Weidner P. Nonconvex separation theorems and some applications in vector optimization. J Optim Theory Appl, 1990, 67: 297--320

[3] Chen G Y, Jahn J. Optimality conditions for set-valued optimization problems. Math Methods Oper Res, 1998, 48: 187--200

[4] Corley H W. Optimality condition for maximizations of set-valued Functions. J Optim Theory Appl, 1988, 58: 1--10

[5] Gong X H, Dong H B, Wang S Y. Optimality conditions for proper efficient solutions of vector set-valued optimization. J Math Anal Appl, 2003, 284: 332--350

[6] Jahn J. Vector Optimization-Theory, Applications and Extensions. Berlin: Springer, 2004

[7] Huang L G. The tangent cones on constraint qualifications in optimization problems. Acta Mathematica Scientia, 2008, 28B(4): 843--850

[8] Jahn J, Rauh R. Contingent epiderivatives and set-valued optimization. Math Methods Oper Res, 1997, 46: 193--211

[9] Jahn J, Khan A A. Generalized contingent epiderivatives in set-valued optimization: optimality conditions. Numer Funct Anal Optim, 2002, 23(7/8): 807--831

[10] Jahn J, Khan A A, Zeilinger P. Second-order optimality conditions in set optimalization. J Optim Theory Appl, 2005, 125: 331--347

[11] Jiménez B, Novo V. Second-order necessary conditions in set constrained differentiable vector optimization. Math Methods Oper Res, 2003, 58: 299--317

[12] Khanh P Q, Tuan N D. Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimization. J Optim Theory Appl, 2008, 139(2): 243--261

[13] Li S J, Chen C R. Higher-order optimality conditions for Henig efficient solutions in set-valued optimization. J Math Anal Appl, 2006, 323: 1184--1200

[14] Li S J, Teo K L, Yang X Q. Higher-order optimality conditions for set-valued optimization. J Optim Theory Appl, 2008, 137(3): 533--553

[15] Li S J, Teo K L, Yang X Q. Higher-order Mond-Weir duality for set-valued optimization. J Comput Appl Math, 2008, 217(2): 339--349

[16] Li S J, Yang X Q, Chen G Y. Nonconvex vector optimization of set-valued mappings. J Math Anal Appl, 2003, 283: 337--350

[17] Sheng B H, Liu S Y. On the generalized Fritz John optimality conditions of vector optimization with set-valued maps under Benson proper efficiency. Applied Mathematics and Mechanics(English Edition), 2002, 23(12): 1444--1451