一类广义不变凸函数和向量变分不等式

数学物理学报 ›› 2017, Vol. 37 ›› Issue (6): 1029-1039.

一类广义不变凸函数和向量变分不等式

李茹¹, 余国林¹, 刘伟¹, 刘三阳²

1 北方民族大学应用数学研究所银川 750021;
2 西安电子科技大学理学院西安 710071

收稿日期:2016-12-03 修回日期:2017-04-17 出版日期:2017-12-26 发布日期:2017-12-26
通讯作者: 余国林,E-mail:guolin_yu@126.com E-mail:guolin_yu@126.com
基金资助:
国家自然科学基金（11361001）和宁夏自然科学基金（NZ14101）

A Class of Generalized Invex Functions and Vector Variational Inequalities

Li Ru¹, Yu Guolin¹, Liu Wei¹, Liu Sanyang²

1 Institute of Applied Mathematics, Beifang University of Nationalities Yinchuan, Ningxia 750021;
2 College of Science, Xidian University, Xi'an 710071

Received:2016-12-03 Revised:2017-04-17 Online:2017-12-26 Published:2017-12-26
Supported by:
Supported by the NSFC(11361001) and the Ningxia Natural Science Foundation (NZ14101)

摘要/Abstract

摘要： 首先在函数不变凸性的基础上，引进了一类广义不变凸函数，称之为（α，ρ，η）-不变凸函数，并给出实例说明了这类广义凸函数的存在性.其次，在（α，ρ，η）-不变凸性假设下，研究了Stampacchia型和Minty型向量变分不等式与多目标规划解之间的关系.最后，利用KKM定理讨论了向量变分不等式解的存在性问题.

关键词: 向量变分不等式, 多目标规划, 不变凸函数, KKM定理

Abstract: In this paper, a new kind of generalized invex functions, termed (α,ρ,η)-invex function, is introduced, and some examples are proposed to illustrate its existence. Under the assumption of (α,ρ,η)-invexity, it presents the close relationship between variational-like inequality and multi-objective programming. By using KKM theorem, the existence results are established for vector variational-like inequalities.

Key words: Vector variational inequalities, Multi-objective programming, Invex functions, KKM theorem

中图分类号:

O221

李茹, 余国林, 刘伟, 刘三阳. 一类广义不变凸函数和向量变分不等式[J]. 数学物理学报, 2017, 37(6): 1029-1039.

Li Ru, Yu Guolin, Liu Wei, Liu Sanyang. A Class of Generalized Invex Functions and Vector Variational Inequalities[J]. Acta mathematica scientia,Series A, 2017, 37(6): 1029-1039.

参考文献

[1] Hanson M A. On sufficienty of the Kuhn-Tucker conditions. Journal of Mathematical Analysis and Applications, 1981, 80:545-550
[2] Jeyakumar V, Mond B. On generalised convex mathematical programming. Journal of the Australian Mathematical Society, 1992, 34:43-53
[3] Emam T. Roughly B-invex multi-objective programming problems. Open Journal of Optimization, 2012, 1:1-7
[4] Ivanov V I. Second-order invex functions in nonlinear programming. Optimization, 2012, 5:489-503
[5] Yang X M, Yang X Q. Vector variational-like inequality with pseudoinvexity. Optimization, 2006, 55(1/2):157-170
[6] Yu G L, Liu S Y. Optimality for (h, φ)-multiobjective programming involving generalized type-I functions. Journal of Global Optimization, 2008, 41:147-161
[7] Giannessi F. On Minty Variational Principle. New Trends in Mathematical Programming. Boston:Kluwer Academic Publisher, 1998
[8] Al-Homidan S, Ansari Q H. Generalized Minty vector variational-like inequalities and vector optimization problems. Journal of Optimization Theory and Applications, 2010, 14:1-11
[9] Rezaie M, Zafarani J. Vector optimization and variational-like inequalities. Journal of Global Optimization, 2009, 43:47-66
[10] Yang X M, Yang X Q, Teo K L. Some remarks on the Minty vector variational inequality. Journal of Optimization Theory and Applications, 2004, 121(1):193-201
[11] Long X J, Peng J W, Wu S Y. Generalized vector variational-like inequalities and nonsmooth vector optimization problems. Optimization, 2012, 61:1075-1086
[12] Ansari Q H, Rezaie M. Existence results for Stampacchia and Minty type vector variational inequalities. Optimization, 2010, 59 (7):1053-1065
[13] Dos Santos Batista L B, Ruiz-Garzónb G, Rojas-Medar M A, Rufián-Lizanad A. Some relations between variational like inequality problems and vector optimization problems in Banach spaces. Computers Mathematics with Applications, 2008, 55:1808-1814
[14] Farajzadeh A P, Lee B S. Vector variational-like inequality problem and vector optimization problem. Applied Mathematics Letters, 2010, 23:48-52
[15] Zeng J, Li S J. On vector variational-like inequalities and set-valued optimization problems. Optimization Letters, 2011, 5(1):55-69
[16] Long X J, Quan J, Wen D J. Proper efficiency for set-valued optimization problems and vector variational-like inequalities. Bulletin of the Korean Mathematical Society, 2013, 50:777-786
[17] Ben-Israel M A, Mond B. What is invexity. Journal of the Australian Mathematical Society, 1986, 28:1-9
[18] Reiland T W. Nonsmooth invexity. Bulletin of the Australian Mathematical Society, 1990, 42:437-446
[19] Ceng L C, Huang S. Existence theorems for generalized vector variational inequalities with a variable ordering relation. Journal of Global Optimization, 2010, 46:521-535