Asplund空间中非凸向量均衡问题近似解的最优性条件

数学物理学报 ›› 2014, Vol. 34 ›› Issue (3): 593-602.

Asplund空间中非凸向量均衡问题近似解的最优性条件

龙宪军

重庆工商大学数学与统计学院重庆 400067

收稿日期:2012-09-18 修回日期:2013-04-16 出版日期:2014-06-25 发布日期:2014-06-25
基金资助:
国家自然科学基金(11001287, 71271226)、重庆市自然科学基金(CSTC 2010BB9254, CSTC 2012jjA00039)和重庆市教委科技研究项目(KJ100711)资助

Optimality Conditions for Approximate Solutions on Nonconvex Vector Equilibrium Problems in Asplund Spaces

LONG Xian-Jun

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 40006

Received:2012-09-18 Revised:2013-04-16 Online:2014-06-25 Published:2014-06-25
Supported by:
国家自然科学基金(11001287, 71271226)、重庆市自然科学基金(CSTC 2010BB9254, CSTC 2012jjA00039)和重庆市教委科技研究项目(KJ100711)资助

摘要/Abstract

摘要：

在Asplund空间中, 研究了非凸向量均衡问题近似解的最优性条件. 借助Mordukhovich次可微概念, 在没有任何凸性条件下获得了向量均衡问题εe -拟弱有效解, εe -拟Henig 有效解, εe -拟全局有效解以及εe -拟有效解的必要最优性条件. 作为它的应用, 还给出了非凸向量优化问题近似解的最优性条件.

关键词: 非凸向量均衡问题, 近似解, 最优性条件, Mordukhovich次可微

Abstract:

The purpose of this paper is to study approximate solutions for the vector equilibrium problem in Asplund spaces without any convexity assumption. We obtain optimality conditions for εe-quasi weakly efficient solutions, εe-quasi Henig efficient solutions, εe-quasi globally efficient solutions and εe-quasi efficient solutions to vector equilibrium problems
by the Mordukhovich subdifferential. As applications of our results, we derive some optimality conditions for nonconvex vector optimization problems.

Key words: Nonvector vector equilibrium problem, Approximate solution, Optimality condition, Mordukhovich subdifferential

中图分类号:

90C26

龙宪军. Asplund空间中非凸向量均衡问题近似解的最优性条件[J]. 数学物理学报, 2014, 34(3): 593-602.

LONG Xian-Jun. Optimality Conditions for Approximate Solutions on Nonconvex Vector Equilibrium Problems in Asplund Spaces[J]. Acta mathematica scientia,Series A, 2014, 34(3): 593-602.

参考文献

[1] Ansari Q H, Konnov I V, Yao J C. On generalized vector equilibriums. Nonlinear Analysis (TMA), 2001, 47: 543--554

[2] Bianchi M, Hadjisavvas N, Schaibles S. Vector equilibrium problems with generalized monotone bifunctions. J Optim Theory Appl, 1997, 92: 527--542

[3] Chen G Y, Huang X X, Yang X Q. Vector Optimization: Set-Valued and Variational Analysis. Berlin, Heidelberg: Springer-Verlag, 2005

[4] Fu J Y. Generalized vector quasi-equilibrium problem. Math Meth Oper Res, 2000, 52: 57--64

[5] Giannessi F. (ed.) Vector Variational Inequilities and Vector Equilibria: Mathematical Theories. Dordrechet: Kluwer, 2000

[6] Long X J, Huang N J, Teo K L. Existence and stability of solutions for generalized strong vector quasi-equilibrium problem. Math Comput Modelling, 2008, 47: 445--451

[7] Long X J, Peng J W. Connectedness and compactness of weak efficient solutions for vector equilibrium problems.
Bull Korean Math Soc, 2011, 48: 1225--1233

[8] Gong X H. Optimality conditions for vector equilibrium problems. J Math Anal Appl, 2008, 342: 1455--1466

[9] Long X J, Huang Y Q, Peng Z Y. Optimality conditions for the Henig efficient solution of vector equilibrium problems with constraints. Optim Lett, 2011, 5: 717--728

[10] Gong X H. Scalarization and optimality conditions for vector equilibrium problems. Nonlinear Anal, 2010, 73: 3598--3612

[11] Mordukhovich B S. Variational Analysis and Generalizd Differentiation. Vol. I: Basic Theory, vol 330. Berlin: Springer, 2006

[12] Gerth(Tammer) C, Weidner P. Nonconvex separation theorems and some applications in vector optimization. J Optim Theory Appl, 1990, 67: 297--320

[13] Loridan P. Necessary conditions for ε-optimality. Math Program Study, 1982, 19: 140--152

[14] Rockafellar R T, Wets R J B. Variational Analysis.Berlin: Springer, 1998

[15] 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

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

[17] Zaffaroni A. Degrees of efficiency and degrees of minimality. SIAM J Optim, 2003, 42: 1071--1086

[18] Liu C P, Lee H W. Lagrange multiplier rules for approximate solutions in vector optimization. J Ind Manag Optim, 2012, 8: 749--764

[19] Durea J, Dutta J, Tammer C. Lagrange multipliers for ε-pareto solutions in vector optimization with nonsolid cones in Banach spaces. J Optim Theory Appl, 2010, 145: 196--211

[20] Gupta D, Mehra A. Two types of approximate saddle points. Numer Funct Anal Optim, 2008, 29: 532--550