## Abstract

This paper deals with convex optimization problems in the face of data uncertainty within the framework of robust optimization. It provides various properties and characterizations of the set of all robust optimal solutions of the problems. In particular, it provides generalizations of the constant subdifferential property as well as the constant Lagrangian property for solution sets of convex programming to robust solution sets of uncertain convex programs. The paper shows also that the robust solution sets of uncertain convex quadratic programs and sum-of-squares convex polynomial programs under some commonly used uncertainty sets of robust optimization can be expressed as conic representable sets. As applications, it derives robust optimal solution set characterizations for uncertain fractional programs. The paper presents several numerical examples illustrating the results.

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## Acknowledgments

The authors are grateful to the two anonymous referees for valuable comments for the revision of the paper. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2005378). The work was partially completed, while he was a visitor at the University of New South Wales, Sydney.

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Communicated by Sándor Zoltán Németh.

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Jeyakumar, V., Lee, G.M. & Li, G. Characterizing Robust Solution Sets of Convex Programs under Data Uncertainty.
*J Optim Theory Appl* **164**, 407–435 (2015). https://doi.org/10.1007/s10957-014-0564-0

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DOI: https://doi.org/10.1007/s10957-014-0564-0

### Keywords

- Convex optimization problems with data uncertainty
- Robust optimization
- Optimal solution set
- Uncertain convex quadratic programs
- Uncertain sum-of-squares convex polynomial programs