On approximate solutions and saddle point theorems for robust convex optimization
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On approximate solutions and saddle point theorems for robust convex optimization Xiang-Kai Sun1
· Kok Lay Teo2,3 · Jing Zeng1 · Xiao-Le Guo1
Received: 10 July 2018 / Accepted: 12 August 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract This paper provides some new results on robust approximate optimal solutions for convex optimization problems with data uncertainty. By using robust optimization approach (worst-case approach), we first establish necessary and sufficient optimality conditions for robust approximate optimal solutions of this uncertain convex optimization problem. Then, we introduce a Wolfe-type robust approximate dual problem and investigate robust approximate duality relations between them. Moreover, we obtain some robust approximate saddle point theorems for this uncertain convex optimization problem. We also show that our results encompass as special cases some optimization problems considered in the recent literature. Keywords Approximate optimal solutions · Robust convex optimization · Saddle point theorems
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Xiang-Kai Sun [email protected] Kok Lay Teo [email protected] Jing Zeng [email protected] Xiao-Le Guo [email protected]
1
Chongqing Key Laboratory of Social Economy and Applied Statistics, College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
2
School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, WA 6845, Australia
3
Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China
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X.-K. Sun et al.
1 Introduction Nowadays, convex optimization problem has been recognized as a powerful modeling tool that deals with many real-world optimization problems such as automatic control systems, estimation and signal processing, data analysis and modeling, etc. In the last decades, many important results have been established for convex optimization problems in the absence of data uncertainty, see, for example, [1–7] and the references therein. Recently, a great deal of attention has been focused on convex optimization problems with data uncertainty. Robust optimization [8,9] has emerged as an effective and fashionable method to deal with uncertain optimization problems. Many researchers have been attracted to work on the real-world applications of robust optimization in engineering, business, and management. See, for example, [10–16] and the references therein. More specifically, Li et al. [10] obtained some robust strong duality results for optimization problems that arise in machine learning problems of data classification. In [11,12], the authors investigated portfolio selection problems through the study of robust multiobjective optimization. By using robust optimization, Li et al. [13] considered a least square semidefinite programming problem under ellipsoidal data uncertainty, and illustrated how it can be applied in robust correlation stress testing where data uncertainty a
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