Approximate Optimality Conditions for Composite Convex Optimization Problems
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Approximate Optimality Conditions for Composite Convex Optimization Problems Xian-Jun Long1 · Xiang-Kai Sun1 · Zai-Yun Peng2
Received: 11 February 2016 / Revised: 1 August 2016 / Accepted: 11 October 2016 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2016
Abstract The purpose of this paper is to study the approximate optimality condition for composite convex optimization problems with a cone-convex system in locally convex spaces, where all functions involved are not necessarily lower semicontinuous. By using the properties of the epigraph of conjugate functions, we introduce a new regularity condition and give its equivalent characterizations. Under this new regularity condition, we derive necessary and sufficient optimality conditions of ε-optimal solutions for the composite convex optimization problem. As applications of our results, we derive approximate optimality conditions to cone-convex optimization problems. Our results extend or cover many known results in the literature.
This research was supported by the National Natural Science Foundation of China (Nos. 11471059, 11301571, and 11301570), the Chongqing Research Program of Basic Research and Frontier Technology (Nos. cstc2014jcyjA00037, cstc2015jcyjB00001, cstc2015jcyjA00025, and cstc2015jcyjA00002), the Education Committee Project Research Foundation of Chongqing (Nos. KJ1400618 and KJ1500626), the Postdoctoral Science Foundation of China (Nos. 2015M580774 and 2016T90837) and the Program for University Innovation Team of Chongqing (CXTDX201601026 and CXTDX201601022).
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Xian-Jun Long [email protected] Xiang-Kai Sun [email protected] Zai-Yun Peng [email protected]
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College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
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College of Mathematics and Statistics, Chongqing JiaoTong University, Chongqing 400074, China
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X.-J. Long et al.
Keywords Composite convex optimization problem · Approximate optimality condition · Generalized regularity condition · ε-Subdifferential Mathematics Subject Classification 90C25 · 90C30 · 90C46
1 Introduction In practical applications, solving an optimization problem is usually to find a point where the minimal or the maximal value a function can take is attained. Unfortunately, it is not always possible, because an optimization problem does not necessarily have an optimal solution. So we are forced sometimes to deal with approximate solutions. Moreover, a lot of solution methods (for example, iterative algorithms or heuristic algorithms) usually find the approximate solution, not an optimal solution. Therefore, the study of approximate solutions of an optimization problem is of great interest from both the theoretical and practical points of view and many authors have turned their attention to this topic; see [1–3] and the references therein. Recently, under the assumptions of lower semi-continuity, Bot et al. [4] obtained a necessary and sufficient condition for
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