一类带多项式约束的不确定凸优化问题的鲁棒可行性半径刻画

Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (5): 1551-1559.

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Characterization of the Radius of the Robust Feasibility for a Class of Uncertain Convex Optimization Problems with Polynomial Constraints

Caiyun Xiao(),Xiangkai Sun*()

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

Received:2021-11-26 Online:2022-10-26 Published:2022-09-30
Contact: Xiangkai Sun E-mail:xcyuncq@163.com;sunxk@ctbu.edu.cn
Supported by:
the NSFC(12001070);the Natural Science Foundation of Chongqing(cstc2020jcyj-msxmX0016);the Science and Technology Research Program of Chongqing Municipal Education Commission(KJZD-K202100803);the Innovation Project of CTBU(yjscxx2022-203-185);the Education Committee Project Foundation of Chongqing for Bayu Young Scholar

Abstract

Abstract:

This paper deals with the lower bound of the radius of the robust feasibility for a class of convex optimization problems with uncertain convex polynomial constraints. Following the idea due to robust optimization, we first introduce the robust counterpart of the uncertain convex optimization problem and give the concept of radius of robust feasibility. By using the so-called epigraphical set and the Minkowski functions generated by the uncertain sets, we obtain the lower bound for the radius of robust feasibility of the uncertain convex optimization. Furthermore, an exact formula for the radius of the robust feasibility for an uncertain optimization problem with SOS-convex polynomial constraints is obtained. Our results extend and improve the corresponding results obtained in [10].

Key words: Polynomial constraints, Robust feasibility, Minkowski functions

CLC Number:

O221.8

Caiyun Xiao,Xiangkai Sun. Characterization of the Radius of the Robust Feasibility for a Class of Uncertain Convex Optimization Problems with Polynomial Constraints[J].Acta mathematica scientia,Series A, 2022, 42(5): 1551-1559.

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References 18

1	Ben-Tal A , Nemirovski A . Robust convex optimization-methodology and applications. Math Program Ser B, 2002, 92, 453- 480 doi: 10.1007/s101070100286
2	Beck A , Ben-Tal A . Duality in robust optimization: primal worst equals dual best. Oper Res Lett, 2009, 37, 1- 6 doi: 10.1016/j.orl.2008.09.010
3	Ben-Tal A , El Ghaoui L , Nemirovski A . Robust Optimization. Princeton(NJ): Princetion University Press, 2009
4	Klamroth K , Köbis E , Schöbel A , Tammer C . A unified approach for different concepts of robustness and stochastic programming via non-linear scalarizing functionals. Optimization, 2013, 65, 649- 671
5	Sun X K , Peng Z Y , Guo X L . Some characterizations of robust optimal solutions for uncertain convex optimization problems. Optim Lett, 2016, 10, 1463- 1478 doi: 10.1007/s11590-015-0946-8
6	孙祥凯. 不确定信息下凸优化问题的鲁棒解刻划. 数学物理学报, 2017, 37A (2): 257- 264
	Sun X K . Characterizations of robust solution for convex optimization problems with data uncertainty. Acta Math Sci, 2017, 37A (2): 257- 264
7	叶冬平, 方东辉. 鲁棒复合优化问题的Lagrange对偶. 数学物理学报, 2020, 40A (4): 1095- 1107 doi: 10.3969/j.issn.1003-3998.2020.04.024
	Ye D P , Fang D H . Lagrange dualities for robust composite optimization problems. Acta Math Sci, 2020, 40A (4): 1095- 1107 doi: 10.3969/j.issn.1003-3998.2020.04.024
8	Goberna M A , Jeyakumar V , Li G , Vicente-Pérez J . Robust solutions of multi-objective linear semi-infinite programs under constraint data uncertainty. SIAM J Optim, 2014, 24, 1402- 1419 doi: 10.1137/130939596
9	Goberna M A , Jeyakumar V , Li G , Vicente-Pérez J . Robust solutions to multi-objective linear programs with uncertain data. European J Oper Res, 2015, 242, 730- 743 doi: 10.1016/j.ejor.2014.10.027
10	Goberna M A , Jeyakumar V , Li G , Linh N . Radius of robust feasibility formulas for classes of convex programs with uncertain polynomial constraints. Oper Res Lett, 2016, 44, 67- 73 doi: 10.1016/j.orl.2015.11.011
11	Chuong T D , Jeyakumar V . An exact formula for radius of robust feasibility of uncertain linear programs. J Optim Theory Appl, 2017, 173, 203- 226 doi: 10.1007/s10957-017-1067-6
12	Chen J , Li J , Li X , Lv Y , Yao J C . Radius of robust feasibility of system of convex inequalities with uncertain data. J Optim Theory Appl, 2020, 184, 384- 399 doi: 10.1007/s10957-019-01607-7
13	Rockafellar R T . Convex Analysis. Princeton(NJ): Princeton University Press, 1970
14	Dinh N , Cánovas M J , López M A . From linear to convex systems: consistency Farkas lemma and applications. J Convex Anal, 2006, 13, 279- 290
15	Schirotzek W . Nonsmooth Analysis. Berlin: Springer, 2006
16	Helton J W , Nie J W . Semidefinite representation of convex sets. Math Program, 2010, 122, 21- 64 doi: 10.1007/s10107-008-0240-y
17	Ahmadi A A , Parrilo P A . A convex polynomial that is not SOS-convex. Math Program, 2012, 135, 275- 292 doi: 10.1007/s10107-011-0457-z
18	Cánovas M J , López M A , Parra J , Toledo F J . Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems. Math Program, 2005, 103A, 95- 126

Characterization of the Radius of the Robust Feasibility for a Class of Uncertain Convex Optimization Problems with Polynomial Constraints

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