鲁棒复合优化问题的Lagrange对偶

数学物理学报 ›› 2020, Vol. 40 ›› Issue (4): 1095-1107.

鲁棒复合优化问题的Lagrange对偶

叶冬平(),方东辉*()

^{吉首大学数学与统计学院湖南吉首 416000}

收稿日期:2019-07-23 出版日期:2020-08-26 发布日期:2020-08-20
通讯作者: 方东辉 E-mail:1441638656@qq.com;dh_fang@jsu.edu.cn
作者简介:叶冬平, E-mail:1441638656@qq.com
基金资助:
国家自然科学基金(11861033);湖南省自然科学基金(2020JJ4494);湖南省教育厅科研项目(17A172);吉首大学校级科研项目(Jdy19001)

Lagrange Dualities for Robust Composite Optimization Problems

Dongping Ye(),Donghui Fang*()

^{College of Mathematics and Statistics, Jishou University, Hunan Jishou 416000}

Received:2019-07-23 Online:2020-08-26 Published:2020-08-20
Contact: Donghui Fang E-mail:1441638656@qq.com;dh_fang@jsu.edu.cn
Supported by:
the NSFC(11861033);the NSF of Hunan Province(2020JJ4494);the Scientific Research Fund of Hunan Provincial Education Department(17A172);the Scientific Research Fund of Jishou University(Jdy19001)

摘要/Abstract

摘要：

利用共轭函数的上图性质，引入两类新的约束规范条件，等价刻画了鲁棒复合优化问题与其对偶问题之间的Lagrange零对偶，强对偶，稳定零对偶及稳定强对偶，推广和改进了前人的相关结论.

关键词: 鲁棒复合优化问题, 约束规范条件, 零对偶, 强对偶

Abstract:

By using the properties of the epigraph of the conjugate functions, two new constraint qualifications are given. Under these constraint qualifications, the zero duality, the strong duality, the stable zero duality and the stable strong duality between robust composite optimization problem and its Lagrange dual problems are established, which extend and improve the corresponding results in the previous papers.

Key words: Robust composite optimization problem, Constraint qualification, Zero duality, Strong duality

中图分类号:

O224

叶冬平,方东辉. 鲁棒复合优化问题的Lagrange对偶[J]. 数学物理学报, 2020, 40(4): 1095-1107.

Dongping Ye,Donghui Fang. Lagrange Dualities for Robust Composite Optimization Problems[J]. Acta mathematica scientia,Series A, 2020, 40(4): 1095-1107.

参考文献

1	Auslender A , Teboulle M . Asymptotic Cones and Functions in Optimization and Variational Inequalities. New York: Springer-Verlag, 2003
2	Auslender A . Existence of optimal solutions and duality results under weak conditions. Math Program Ser A, 2000, 88, 45- 59
3	Bo? R I , Grad S M , Wanka G . Generalized Moreau-Rockafellar results for composed convex functions. Optim, 2009, 58, 917- 933
4	Bo? R I , Jeyakumar V , Li G Y . Robust duality in parametric convex optimization. Set-Valued Var Anal, 2013, 21, 177- 189
5	Ban L , Song W . Duality gap of the conic convex constrained optimization problems in normed spaces. Math Program Ser A, 2009, 119, 195- 214
6	Fang D H , Ansari Q H , Zhao X P . Constraint qualifications and zero duality gap properties in conical programming involving composite functions. J Nonlinear Convex Anal, 2018, 19, 53- 69
7	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
8	Fang D H , Li C , Ng K F . Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming. Nonlinear Anal, 2010, 73, 1143- 1159
9	Fang D H , Li C , Yao J C . Stable Lagrange dualities for robust conical programming. J Nonlinear Convex Anal, 2015, 16, 2141- 2158
10	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
11	胡玲莉, 方东辉. 带锥约束的复合优化问题的最优性条件. 数学物理学报, 2018, 38A (6): 1112- 1121
11	Hu L L , Fang D H . Optimality conditions for composite optimization problems with conical constraints. Acta Math Sci, 2018, 38A (6): 1112- 1121
12	Jeyakumar V , Li G Y . New dual constraint qualifications characterizing zero duality gaps of convex programs and semidefinite programs. Nonlinear Anal, 2009, 71, 2239- 2249
13	Jeyakumar V , Li G Y . Strong duality in robust convex programming:complete characterizations. SIAM J Optim, 2010, 20, 3384- 3407
14	Jeyakumar V , Li G Y , Wang J H . Some robust convex programs without a duality gap. J convex Anal, 2013, 20, 377- 394
15	Jeyakumar V , Wolkowicz H . Zero duality gaps in infinite-dimensional programming. J Optim Theory Appl, 1990, 67, 87- 108
16	Li G Y , Jeyakumar V , Lee G M . Robust conjugate duality for convex optimization under uncertainty with application to data classification. Nonlinear Anal, 2011, 74, 2327- 2341
17	Long X J , Sun X K , Peng Z Y . Approximate optimality conditions for composite convex optimization problems. J Oper Res Soc China, 2017, 5, 469- 485
18	孙祥凯. 复合凸优化问题全对偶性的等价刻画. 吉林大学学报(理学版), 2015, 53, 33- 36
18	Sun X K . Some characterizations of total duality for a composed convex optimization. Journal of Jilin University(Science Edition), 2015, 53, 33- 36
19	Zǎlinescu C . Convex Analysis in General Vector Spaces. New Jersey: World Scientific, 2002
20	赵丹, 孙祥凯. 复合凸优化问题的稳定强对偶. 吉林大学学报(理学版), 2013, 51, 441- 443
20	Zhao D , Sun X K . Stable strong duality for a composed convex optimization problem. Journal of Jilin University (Science Edition), 2013, 51, 441- 443

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[4]	方东辉,刘伟玲. 带复合函数的分式优化问题的Farkas引理[J]. 数学物理学报, 2018, 38(5): 842-854.
[5]	陈哲. 向量优化中广义增广拉格朗日对偶理论及应用[J]. 数学物理学报, 2008, 28(3): 570-577.
[6]	李师正, 张玉芬. 有效解适合鞍点准则的条件[J]. 数学物理学报, 1996, 16(3): 310-315.