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Some characterizations of robust optimal solutions for uncertain convex optimization problems

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Abstract

In this paper, we consider robust optimal solutions for a convex optimization problem in the face of data uncertainty both in the objective and constraints. By using the properties of the subdifferential sum formulae, we first introduce a robust-type subdifferential constraint qualification, and then obtain some completely characterizations of the robust optimal solution of this uncertain convex optimization problem. We also investigate Wolfe type robust duality between the uncertain convex optimization problem and its uncertain dual problem by proving duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. Moreover, we show that our results encompass as special cases some optimization problems considered in the recent literature.

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Acknowledgments

The authors would like to thank the two anonymous referees for valuable comments and suggestions, which helped to improve the paper. This research was supported by the National Natural Science Foundation of China (11301570 and 11301571), the Basic and Advanced Research Project of CQ CSTC (cstc2015jcyjA00002, cstc2015jcyjA00025 and cstc2015jcyjA00038), the Education Committee Project Research Foundation of Chongqing (KJ1500626), the China Postdoctoral Science Foundation funded project (2014T70850 ), and the Chongqing Postdoctoral Science Foundation funded project (xm2014026).

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Correspondence to Xiang-Kai Sun.

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Sun, XK., Peng, ZY. & Guo, XL. Some characterizations of robust optimal solutions for uncertain convex optimization problems. Optim Lett 10, 1463–1478 (2016). https://doi.org/10.1007/s11590-015-0946-8

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