Recession Functions and Unboundedness of Functions

LI Mei-shu; GAO Ying

doi:10.21656/1000-0887.370307

Volume 38 Issue 10

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2017 > 38(10): 1187-1194

LI Mei-shu, GAO Ying. Recession Functions and Unboundedness of Functions[J]. Applied Mathematics and Mechanics, 2017, 38(10): 1187-1194. doi: 10.21656/1000-0887.370307

Citation:

LI Mei-shu, GAO Ying. Recession Functions and Unboundedness of Functions[J]. Applied Mathematics and Mechanics, 2017, 38(10): 1187-1194. doi: 10.21656/1000-0887.370307

Citation:

LI Mei-shu, GAO Ying. Recession Functions and Unboundedness of Functions[J]. Applied Mathematics and Mechanics, 2017, 38(10): 1187-1194. doi: 10.21656/1000-0887.370307

PDF( 798 KB)

Recession Functions and Unboundedness of Functions

doi: 10.21656/1000-0887.370307

School of Mathematical Sciences, Chongqing Normal University, Chongqing 400047, P.R.China

Funds: The National Natural Science Foundation of China(11201511；11771064)

Received Date: 2016-10-11
Rev Recd Date: 2017-09-12
Publish Date: 2017-10-15

Abstract

Abstract

The unboundedness of functions was investigated with the recession cones and recession functions. Firstly, the mean value theorem and recession cones were used to characterize the subdifferentials of convex functions on condition of nondifferentiability. Based on the above results, the necessary and sufficient conditions for recession functions under the subdifferentiable assumption were given. Secondly, the convexity was generalized to E-convex functions, and the unbounded feature of E-convex functions was studied by means of recession functions under the special sublinear assumption. Finally, an example was given to indicate that these results can not be extended to quasiconvex functions.
- recession cone,
- recession function,
- generalized convex function,
- subdifferential,
- unboundedness

FullText(HTML)

References(15)

References

[1]	Rockafellar R T. Convex Analysis [M]. Princeton N J: Princeton University Press, 1970.
[2]	黄学祥. 广义回收锥与广义回收函数[J]. 湘潭大学自然科学学报, 1990(4): 17-22.(HUANG Xue-xiang. Generalized recovery cone and generalized recovery function[J]. Natural Science Journal of Xiangtan University,1990(4): 17-22.(in Chinese))
[3]	唐莉萍, 李飞, 赵克全, 等. 关于向量优化问题的Δ函数标量化刻画的某些注记[J]. 应用数学和力学, 2015,36(10): 1095-1106.(TANG Li-ping, LI Fei, ZHAO Ke-quan, et al. About the vector optimization problems Δ function standard quantitative characterization of some note[J]. Applied Mathematics and Mechanics,2015,36 (10) : 1095-1106.(in Chinese))
[4]	赵勇, 彭再云, 张石生. 向量优化问题有效点集的稳定性[J]. 应用数学和力学, 2013,34(6): 643-650.(ZHAO Yong, PENG Zai-yun, ZHANG Shi-sheng. Stability of effective point set for vector optimization[J]. Applied Mathematics and Mechanics,2013,34(6): 643-650.(in Chinese))
[5]	李小燕, 高英. 多目标优化问题proximal真有效解的最优性条件[J]. 应用数学和力学, 2015,36(6): 668-676.(LI Xiao-yan, GAO Ying. The optimal conditions for the effective solution of the multi-objective optimization problem[J]. Applied Mathematics and Mechanics,2015,36(6): 668-676.(in Chinese))
[6]	Obuchowska W T. On the minimizing trajectory of convex functions with unbounded level sets[J]. Computational Optimization Applications,2003,27(1): 37-52.
[7]	Obuchowska W T, Murty K G. Cone of recession and unboundedness of convex functions[J]. European Journal of Operational Research,2001,133(2): 409-415.
[8]	Luc D T. Recession cones and the domination property in vector optimization[J]. Mathematical Programming,1991,49(1): 113-122.
[9]	Obuchowska W T. Unboundedness in reverse convex and concave integer programming[J]. Mathematical Methods of Operations Research,2010,72(2): 187-204.
[10]	Deng S. Boundedness and nonemptiness of the efficient solution sets in multiobjective optimization[J]. Journal of Optimization Theory Applications,2010,144(1): 29-42.
[11]	CHEN Zhe. Asymptotic analysis in convex composite multiobjective optimization problems[J]. Journal of Global Optimization,2013,55(3): 507-520.
[12]	宁刚. E-凸函数的若干特征[J]. 运筹学学报, 2007,11(1): 121-126.(NING Gang. A number of characteristics of the E-convex function[J]. Journal of Operational Research,2007,11(1): 121-126.(in Chinese))
[13]	史树中. 非光滑分析[J]. 数学进展, 1986,15(1): 9-21.(SHI Shu-zhong. Nonsmooth analysis[J]. Mathematical Progress,1986,15(1): 9-21.(in Chinese))
[14]	杨新民, 戎卫东. 广义凸性及其应用[M]. 北京: 科学出版社, 2016.(YANG Xin-min, RONG Wei-dong. Generalized Convexity and Its Application [M]. Beijing: Science Press, 2016.(in Chinese))
[15]	应玫茜, 魏权龄. 非线性规划及其理论[M]. 北京: 中国人民大学出版社, 1994.(YING Mei-qian, WEI Quan-ling. The Theory of Nonlinear Programming and Its Theory [M]. Beijing: Chinese People’s University Press, 1994.(in Chinese))