集值优化问题严格有效解的一、二阶最优性条件

数学物理学报 ›› 2014, Vol. 34 ›› Issue (1): 193-206.

集值优化问题严格有效解的一、二阶最优性条件

柴艳飞|刘三阳|李军

西安电子科技大学数学与统计学院系西安 710071；西华师范大学数学与信息学院四川南充 637002

收稿日期:2012-04-07 修回日期:2013-08-17 出版日期:2014-02-25 发布日期:2014-02-25
通讯作者: 西安电子科技大学数学与统计学院系ad 西安 710071 E-mail:chyf_0923@163.com
基金资助:
国家自然科学基金(61373174, 11371015)、教育部科学技术重点项目(211163)和四川省青年科技基金(2012JQ0032)资助.

First and Second Order Optimality Conditions for Strict Minimality in Set-valued Optimization

LIU San-Yang, LI Jun

Department of Applied Mathematics, Xidian University, Xi'an |710071； School of Mathematics and Information, China West Normal University, Sichuan Nanchong |637002

Received:2012-04-07 Revised:2013-08-17 Online:2014-02-25 Published:2014-02-25
Supported by:
国家自然科学基金(61373174, 11371015)、教育部科学技术重点项目(211163)和四川省青年科技基金(2012JQ0032)资助.

摘要/Abstract

摘要：

首先定义了集值优化问题的m阶局部严格有效解并在赋范空间中研究了解的一些性质. 在一定条件下, 利用Dini 导算子和支撑函数确立了≥1阶严格有效解存在的充分必要条件.

关键词: 集值优化, 一、二阶最有性条件, 支撑函数, 严格最优解, Dini 导算子, 切锥

Abstract:

In this paper, the notion of local strict minimum of order m (m≥1) for vector optimization problems is extended
from single-valued maps to set-valued maps. Some properties and characterizations are then investigated in normed spaces. Furthermore, necessary and sufficient conditions for strict minimum of orders m≥1 are established by using Dini derivatives and support functions.

Key words: Set-valued optimization, First and second order optimality, Support function, Strict minimum, Dini derivative, Tangent cone

中图分类号:

90C25

柴艳飞, 刘三阳, 李军. 集值优化问题严格有效解的一、二阶最优性条件[J]. 数学物理学报, 2014, 34(1): 193-206.

CHAI Yan-Fei, LIU San-Yang, LI Jun. First and Second Order Optimality Conditions for Strict Minimality in Set-valued Optimization[J]. Acta mathematica scientia,Series A, 2014, 34(1): 193-206.

参考文献

[1] Auslender A. Stability in mathematical programming with nondifferentiable data. SIAM J Control Optim, 1984, 22:
239--254

[2] Bednarczuk E M, Song W. Contingent epiderivative and its application to set-valued optimization. Control Cybernet, 1998, 27: 375--386

[3] Bigi G, Castellani M. K-epiderivatives for set-valued functions and optimization. Math Meth Oper Ress, 2002, 55: 401--412

[4] Chen G Y, Huang X X, Yang X Q. Vector Optimization: Set-valued and Variational Analysis. Lecture Notes in
Economics and Mathematical Systems 541. Berlin: Springer-Verlag, 2005

[5] Corley H W. Existence and Lagrangian duality for maximizations of set-valued functions. J Optim Theory Appl, 1987, 54: 489--501

[6] Corley H W. Optimality conditions for maximization of set-valued functions. J Optim Theory Appl, 1988, 58: 1--10

[7] Crespi G P, Ginchev I, Rocca M. First-order optimality conditions in set-valued optimization. Math Meth Oper Res, 2006, 63: 87--106

[8] Hiriart-Urruty J B. Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math Oper Res, 1979, 4: 79--97

[9] Huang L R. Separate necessary and sufficient conditions for the local minimum of a function. J Optim Theory Appl, 2005, 125: 241--246

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

[11] Jim\'{e}nez B. Strict efficiency in vector optimization. J Math Anal Appl, 2002, 265: 264--284

[12] Jim\'{e}nez B, Novo V. First and second order sufficient conditions for strict minimality in nonsmooth vector optimization. J Math Anal Appl, 2003, 284: 496--510

[13] Jim\'{e}nez B, Novo V. Second order necessary conditions in set constrained differentiable vector optimization. Math Meth Oper Res, 2003, 58: 299--317

[14] Jim\'{e}nez B, Novo V. First and second order sufficient conditions for strict minimality in multiobjective programming. Numer Funct Anal Optim, 2002, 23: 303--322

[15] Luc D T. Theory of Vector Optimization. Berlin: Springer-Verlag, 1989

[16] Penot J P. Differentiability of relations and differential stability of perturbed optimization problems. SIAM J Control Optim, 1984, 22: 529--551

[17] Studniarski M. Necessary and sufficient conditions for isolated local minima of nonsmooth function. SIAM J Control
Optim, 1986, 24: 1044--1049

[18] Ward D E. Characterizations of strict local minima and necessary conditions for weak sharp minima. J Optim Theoty
Appl, 1994, 80: 551--571