不确定凸优化问题鲁棒近似解的最优性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (4): 1007-1017.

不确定凸优化问题鲁棒近似解的最优性

李萌(),余国林*()

^{北方民族大学应用数学研究所银川 750021}

收稿日期:2019-03-20 出版日期:2020-08-26 发布日期:2020-08-20
通讯作者: 余国林 E-mail:601575330@qq.com;guolin_yu@126.com
作者简介:李萌, E-mail:601575330@qq.com
基金资助:
国家自然科学基金(11861002);北方民族大学非线性分析与金融优化创新团队和北方民族大学重大专项(ZDZX201804)

Optimality of Robust Approximation Solutions for Uncertain Convex Optimization Problems

Meng Li(),Guolin Yu*()

^{Institute of Applied Mathematics, North Minzu University, Yinchuan 750021}

Received:2019-03-20 Online:2020-08-26 Published:2020-08-20
Contact: Guolin Yu E-mail:601575330@qq.com;guolin_yu@126.com
Supported by:
the NSFC(11861002);the Nonlinear Analysis and Financial Optimization Research Center of North Minzu University and the Key Project of North Minzu University(ZDZX201804)

摘要/Abstract

摘要：

该文研究一类目标和约束函数均带有不确定信息的凸优化问题的鲁棒近似解.首先，在闭凸锥约束品性假设下，得到了该不确定优化问题关于近似解的最优性条件.然后，引入所研究不确定优化问题的近似鞍点的概念，并给出了近似解的鞍点刻划.

关键词: 鲁棒凸优化, 近似解, 近似最优性条件, 近似鞍点

Abstract:

This paper is devoted to study the optimality conditions of robust approximate solutions for a convex optimization problem, in which the objective and constraint functions are carried with uncertain data. First of all, under the assumption of a closed convex cone constraint qualification for the constraint functions, the optimality conditions of approximate solutions for the concerned uncertain optimization problem are presented. Then, the concept of robust approximate saddle point is introduced, and the characterization of robust approximate solutions are proposed by saddle points.

Key words: Robust convex optimization, Approximate solution, Approximate optimality condition, Approximate saddle point

中图分类号:

O224

李萌,余国林. 不确定凸优化问题鲁棒近似解的最优性[J]. 数学物理学报, 2020, 40(4): 1007-1017.

Meng Li,Guolin Yu. Optimality of Robust Approximation Solutions for Uncertain Convex Optimization Problems[J]. Acta mathematica scientia,Series A, 2020, 40(4): 1007-1017.

参考文献 14

1	Soyster A L . Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research, 1973, 21 (5): 1154- 1157
2	Ben-Tal A , Bhadra S , Bhattacharyya C , et al. Efficient methods for robust classification under uncertainty in kernel matrices. Journal of Machine Learning Research, 2012, 13 (1): 2923- 2954
3	Ben-Tal A , Boyd S , Nemirovski A . Extending scope of robust optimization:comprehensive robust counterparts of uncertain problems. Mathematical Programming, 2006, 107 (1/2): 63- 89
4	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, 37 (2): 257- 264 doi: 10.3969/j.issn.1003-3998.2017.02.005
	Sun X K . Characterizations of robust solution for convex optimization problems with data uncertainty. Acta Math Sci, 2017, 37 (2): 257- 264 doi: 10.3969/j.issn.1003-3998.2017.02.005
5	Sun X K , Peng Z Y , Guo X L . Some characterizations of robust optimal solutions for uncertain convex optimization problems. Optimization Letters, 2015, 10 (7): 1- 16
6	Lee J H , Jiao L . On quasi Q-solution for robust convex optimization problems. Optimization Letters, 2016, 11, 1- 14
7	Sun X K , Li X B , Long X J , et al. On robust approximate optimal solutions for uncertain convex optimization and applications to multi-objective optimization. Pacific Journal of Optimization, 2017, 13 (4): 621- 643
8	Kim M H . Duality theorem and vector saddle point theorem for robust multiobjective optimization problems. Communications of the Korean Mathematical Society, 2013, 28 (3): 597- 602 doi: 10.4134/CKMS.2013.28.3.597
9	Jeyakumar V . Asymptotic dual conditions characterizing optimality for convex programs. Optim Theory Appl, 1997, 93, 153- 165 doi: 10.1023/A:1022606002804
10	Bot R I . Conjugate duality in convex optimization. Lecture Notes in Economics Mathematical Systems, 2010, 637 (1): 31- 41
11	Jeyakumar V , Li G Y . Strong duality in robust convex programming:complete characterizations. SIAM J Optim, 2010, 20, 3384- 3407 doi: 10.1137/100791841
12	Jeyakumar V , Lee G M , Dinh N . New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM Journal on Control and Optimization, 2003, 14, 534- 547 doi: 10.1137/S1052623402417699
13	XU Z K . On multiple-criteria fractional programming problems. Journal of Systems Science and Mathematical Sciences, 1994, 14 (3): 199- 208
14	李茹, 余国林, 刘伟, 刘三阳. 一类广义不变凸函数和向量变分不等式. 数学物理学报, 2017, 37A (6): 1029- 1039 doi: 10.3969/j.issn.1003-3998.2017.06.003
	Li R , Yu G L , Liu W , Liu S Y . A class of generalized invex functions and vector variational inequalities. Acta Math Sci, 2017, 37A (6): 1029- 1039 doi: 10.3969/j.issn.1003-3998.2017.06.003

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[2]	武宝亭孙彦平. 一类强非线性偏微分方程组初边值问题之逆算符解法新探[J]. 数学物理学报, 1999, 19(3): 347-355.
[3]	闫玉斌, 李春利. 第一类不适定算子方程的解的表示[J]. 数学物理学报, 1998, 18(3): 264-269.