一类带DC函数的分式优化的Farkas引理刻画

Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (3): 827-836.

Previous Articles Next Articles

Characterizations of Farkas Lemmas for a Class of Fractional Optimization with DC Functions

Xinyi Feng(),Xiangkai Sun*()

^{Chongqing Key Laboratory of Social Economy and Applied Statistics, College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067}

Received:2020-06-14 Online:2021-06-26 Published:2021-06-09
Contact: Xiangkai Sun E-mail:1518684363@qq.com;sxkcqu@163.com
Supported by:
the NSF of Chongqing(cstc2020jcyj-msxmX0016);the Open Research Platform of CTBU(KFJJ2019097);the Project of CTBU(ZDPTTD201908);the Education Committee Project Foundation of Chongqing for Bayu Young Scholar

Abstract

Abstract:

This paper deals with some new Farkas lemmas for a class of constraint fractional optimization with DC functions(the difference of convex functions). Following the idea due to Dinkelbach, we first associate the fractional optimization with a DC optimization problem. Then, by using the epigraph technique of the conjugate function, we introduce some new regularity conditions and establish the duality between the DC optimization problem and its Fenchel-Lagrange dual problem. Finally, we obtain some new Farkas lemmas for the fractional optimization problem. Furthermore, we also show that the results obtained in this paper extend and improve the corresponding results in the literature.

Key words: Fractional optimization, Regularity conditions, Farkas lemmas

CLC Number:

O221.2

Xinyi Feng,Xiangkai Sun. Characterizations of Farkas Lemmas for a Class of Fractional Optimization with DC Functions[J].Acta mathematica scientia,Series A, 2021, 41(3): 827-836.

Trendmd

References 16

1	Boţ R I , Wanka G . Farkas-type results with conjugate functions. SIAM J Optim, 2005, 15, 540- 554 doi: 10.1137/030602332
2	Boţ R I , Hodrea I B , Wanka G . Farkas-type results for fractional programming problems. Nonlinear Anal, 2007, 67, 1690- 1703 doi: 10.1016/j.na.2006.07.041
3	Dinh N , Goberna M A , López M A , Son T Q . New Farkas-type constraint qualifications in convex infinite programming. ESAIM Control Optim Calc Var, 2007, 13, 580- 597 doi: 10.1051/cocv:2007027
4	Fang D H , Li C , Ng K F . Constraint qualifications for extended Farkas's lemmas and Lagrangian dualities in convex infinite programming. SIAM J Optim, 2009, 20, 1311- 1332
5	Dinh N , Jeyakumar V . Farkas's lemma: three decades of generalizations for mathematical optimization. Top, 2014, 22, 1- 22 doi: 10.1007/s11750-014-0319-y
6	Zhang X H , Cheng C Z . Some Farkas-type results for fractional programming with DC functions. Nonlinear Anal Real World Appl, 2009, 10, 1679- 1690 doi: 10.1016/j.nonrwa.2008.02.006
7	Sun X K . Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results in DC infinite programming. J Math Anal Appl, 2014, 414, 590- 611 doi: 10.1016/j.jmaa.2014.01.033
8	Sun X K , Chai Y , Zeng J . Farkas-type results for constrained fractional programming with DC functions. Optim Lett, 2014, 8, 2299- 2313 doi: 10.1007/s11590-014-0737-7
9	Fang D H , Gong X . Extended Farkas lemma and strong duality for composite optimization problem with DC functions. Optimization, 2017, 66, 179- 196 doi: 10.1080/02331934.2016.1266628
10	方东辉, 刘伟玲. 带复合函数的分式优化问题的Farkas引理. 数学物理学报, 2018, A (5): 842- 854 doi: 10.3969/j.issn.1003-3998.2018.05.002
	Fang D H , Liu W L . The farkas lemmas for fractional optimization problem with composite functions. Acta Math Sci, 2018, 38A (5): 842- 854 doi: 10.3969/j.issn.1003-3998.2018.05.002
11	Fang D H , Zhang Y . Extended Farkas's lemmas and strong dualities for conic programming involving composite functions. J Optim Theory Appl, 2018, 176, 351- 376 doi: 10.1007/s10957-018-1219-3
12	Sun X K , Long X J , Tang L P . Regularity conditions and Farkas-type results for systems with fractional functions. RAIRO-Oper Res, 2020, 54 (5): 1369- 1384 doi: 10.1051/ro/2019070
13	Dinkelbach W . On nonlinear fractional programming. Manag Sci, 1967, 13, 492- 498 doi: 10.1287/mnsc.13.7.492
14	Zǎlinescu C. Convex Analysis in General Vector Spaces. New Jersey: World Scientific, 2002
15	Rockafellar R T . Convex Analysis. Princeton: Princeton University Press, 1970
16	方东辉, 田利萍, 王仙云. 复合凸优化问题的Fenchel-Lagrang强对偶之研究. 数学物理学报, 2020, 40A (1): 20- 30 doi: 10.3969/j.issn.1003-3998.2020.01.003
	Fang D H , Tian L P , Wang X Y . Strong Fenchel-Lagrange duality for convex optimization problems with composite function. Acta Math Sci, 2020, 40A (1): 20- 30 doi: 10.3969/j.issn.1003-3998.2020.01.003

Characterizations of Farkas Lemmas for a Class of Fractional Optimization with DC Functions

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 16

Related Articles 1

Metrics

Comments

Recommended 10