Abstract
In this article, a mathematical programming problem under affinely parameterized uncertain data with multiple objective functions given by SOS-convex polynomials, denoting by (UMP), is considered; moreover, its robust counterpart, denoting by (RMP), is proposed by following the robust optimization approach (worst-case approach). Then, by employing the well-known \(\epsilon \)-constraint method (a scalarization technique), we substitute (RMP) by a class of scalar problems. Under some suitable conditions, a zero duality gap result, between each scalar problem and its relaxation problems, is established; moreover, the relationship of their solutions is also discussed. As a consequence, we observe that finding robust efficient solutions to (UMP) is tractable by such a scalarization method. Finally, a nontrivial numerical example is designed to show how to find robust efficient solutions to (UMP) by applying our results.
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The authors would like to express their sincere thanks to anonymous referees for their very helpful and valuable suggestions and comments for the paper.
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The first author was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2017R1A5A1015722). The second author was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2018R1C1B6001842).
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Jiao, L., Lee, J.H. Finding efficient solutions in robust multiple objective optimization with SOS-convex polynomial data. Ann Oper Res 296, 803–820 (2021). https://doi.org/10.1007/s10479-019-03216-z
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DOI: https://doi.org/10.1007/s10479-019-03216-z
Keywords
- Multiobjective optimization
- Robust optimization
- Semidefinite programming relaxations
- SOS-convex polynomials