关于向量优化问题的Δ函数标量化刻画的某些注记

唐莉萍; 李飞; 赵克全; 杨新民

doi:10.3879/j.issn.1000-0887.2015.10.009

关于向量优化问题的Δ函数标量化刻画的某些注记

doi: 10.3879/j.issn.1000-0887.2015.10.009

¹重庆工商大学数学与统计学院, 重庆 400067;²内蒙古大学数学科学学院,呼和浩特 010021;³重庆师范大学数学科学学院, 重庆 400047

基金项目: 国家自然科学基金(重点项目)(11431004)；国家自然科学基金（面上项目）(11271391；11201511；11301574)；重庆市科委项目(cstc2014pt-sy00001)

详细信息

作者简介:
唐莉萍（1985—），女，四川资中人，博士生（通讯作者. E-mail: tanglipings@163.com）;李飞（1981—），男，内蒙古巴彦淖儿人，讲师，博士生（E-mail: lifeimath@163.com）;赵克全（1979—），男，四川南充人，副教授，博士（E-mail: kequanz@163.com）;杨新民（1960—），男，四川泸州人，教授，博士生导师（E-mail: xmyang@cqnu.edu.cn）.

中图分类号: O221.6
计量
- 文章访问数: 1951
- HTML全文浏览量: 209
- PDF下载量: 922
- 被引次数: 0
出版历程
- 收稿日期: 2015-01-14
- 修回日期: 2015-08-13
- 刊出日期: 2015-10-15

Some Notes on the Scalarization of Function Δ for Vector Optimization Problems

¹College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, P.R.China;²School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, P.R.China;³School of Mathematical Sciences, Chongqing Normal University, Chongqing 400047, P.R.China

Funds: The National Natural Science Foundation of China(Key Program)(11431004);The National Natural Science Foundation of China（General Program）(11271391;11201511;11301574)

摘要

摘要: 近期，夏远梅等(重庆师范大学（自然科学版）,2015,32(1):12-15)利用Δ函数通过非线性标量化方法研究了向量优化问题的?-真有效解并举例说明了主要结果.笔者指出：其定理1是Gao等(Journal of Industrial and Management Optimization,2011,7（2）: 483-496)建立的定理4.6(i)的特例；其定理2的证明存在不足.通过研究一般的(C,ε)-真有效解的Δ函数非线性标量化，给出了定理2的严谨证明.最后，在?-真有效解存在的情况下举例说明了主要结果.
- 向量优化 /
- 近似真有效解 /
- Δ函数 /
- 非线性标量化
Abstract: Recently，Xia Yuan-mei,et al.(Journal of Chongqing Normal University(Natural Science), 2015,32(1): 12-15) studied the ?-properly efficient solutions to vector optimization problems via scalar function Δ in terms of the nonlinear scalarization method, and gave some examples to illustrate their results. It was point out here that theorem 1 established by Xia Yuan-mei,et al. was a special case of Theorem 4.6(i) obtained by Gao, et al.(Journal of Industrial and Management Optimization,2011,7（2）: 483-496), and the proof of Theorem 2 given by Xia Yuan-mei,et al. had some deficiency. Through investigation the nonlinear scalarization of function Δ for the (C, ε)-properly efficient solutions, theorem 2 obtained by Xia Yuan-mei, et al. was proved again rigorously. In the end, some examples in which ?-properly efficient solutions did exist, were given to illustrate the main results.
- vector optimization /
- approximate properly efficient solution /
- function Δ /
- nonlinear scalarization

HTML全文

参考文献(23)

[1]	Kutateladze S S. Convex ε-programming[J].Soviet Mathematics Doklady,1979,20: 390-393.
[2]	夏远梅, 张万里, 赵克全. ε-真有效解的非线性标量化[J]. 重庆师范大学学报(自然科学版), 2015,32(1): 12-15.(XIA Yuan-mei, ZHANG Wang-li, ZHAO Ke-quan. Nonlinear scalarization of ε-properly efficient solutions[J].Journal of Chongqing Normal University (Natural Science),2015,32(1): 12-15.(in Chinese))
[3]	RONG Wei-dong, MA Yi. Connectedness of ε-super efficient solution set of vector optimization problems with set-valued maps[J].Or Transactions,2000,4(4): 21-32.
[4]	Eichfelder G.Adaptive Scalarization Methods in Multiobjective Optimization [M]. Heidelberg, Berlin: Springer-Verlag, 2008.
[5]	Jahn J.Vector Optimization: Theory, Applications, and Extensions [M]. Berlin: Springer, 2011.
[6]	Soleimani-Damaneh M. An optimization modelling for string selection in molecular biology using Pareto optimality[J].Applied Mathematical Modelling,2011,35(8): 3887-3892.
[7]	Soleimani-Damaneh M. On some multiobjective optimization problems arising in biology[J].International Journal of Computer Mathematics,2011,88(6): 1103-1119.
[8]	Engau A, Wiecek M M. Generating ε-efficient solutions in multiobjective programming[J].European Journal of Operational Research,2007,177(3): 1566-1579.
[9]	Khoshkhabar-Amiranloo S, Soleimani-Damaneh M. Scalarization of set-valued optimization problems and variational inequalities in topological vector spaces[J].Nonlinear Analysis,2012,75(3): 1429-1440.
[10]	Son T Q, Strodiot J J, Nguyen V H. ε-optimality and ε-Lagrangian duality for a nonconvex programming problem with an infinite number of constraints[J].Journal of Optimization Theory and Applications,2009,141(2): 389-409.
[11]	Hiriart-Urruty J B. New concepts in nondifferentiable programming[J].Mémoires de la Société Mathématique de France,1979,60: 57-85.
[12]	Hiriart-Urruty J B. Tangent cones, generalized gradients and mathematical programming in Banach spaces[J].Mathematics of Operations Research,1979,4(1): 79-97.
[13]	Gerth C, Weidner P. Nonconvex separation theorems and some applications in vector optimization[J].Journal of Optimization Theory and Applications,1990,67(2): 297-320.
[14]	Gpfert A, Riahi H, Tammer C, Zalinescu C.Variational Methods in Partially Ordered Spaces [M]. New York: Springer-Verlag, 2003.
[15]	Zaffaroni A. Degrees of efficiency and degrees of minimality[J].SIAM Journal on Control and Optimization,2003,42(3): 1071-1086.
[16]	Loridan P. -solutions in vector minimization problems[J].Journal of Optimization Theory and Applications,1984,43(2): 265-276.
[17]	White D J. Epsilon effieiency[J].Journal of Optimization Theory and Applications,1986,49(2): 319-337.
[18]	Gutiérrez C, Jiménez B, Novo V. A unified approach and optimality conditions for approximate solutions of vector optimization problems[J].SIAM Journal on Optimization,2006,17(3): 688-710.
[19]	Helbig S, Pateva D. On several concepts for ε-efficiency[J].Operations Research Spektrum,1994,16(3): 179-186.
[20]	Gutiérrez C, Jiménez B, Novo V. Optimality conditions via scalarization for a new ε-efficiency concept in vector optimization problems[J].European Journal of Operational Research,2010,201(1): 11-22.
[21]	Ha T X D. The Ekeland variational principle for Henig proper minimizers and super minimizers[J].Journal of Mathematical Analysis and Applications,2010,364(1): 156-170.
[22]	Durea M, Dutta J, Tammer C. Lagrange multipliers for ε-Pareto solutions in vector optimization with nonsolid cones in Banach spaces[J].Journal of Optimization Theory and Applications,2010,145(1): 196-211.
[23]	Gao Y, Yang X M, Teo K L. Optimality conditions for approximate solutions of vector optimization problems[J].Journal of Industrial and Management Optimization,2011,7(2): 483-496.