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On minimizing difference of a SOS-convex polynomial and a support function over a SOS-concave matrix polynomial constraint

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Abstract

In this paper, we establish tractable sum of squares characterizations of the containment of a convex set, defined by a SOS-concave matrix inequality, in a non-convex set, defined by difference of a SOS-convex polynomial and a support function, with Slater’s condition. Using our set containment characterization, we derive a zero duality gap result for a DC optimization problem with a SOS-convex polynomial and a support function, its sum of squares polynomial relaxation dual problem, the semidefinite representation of this dual problem, and the dual problem of the semidefinite programs. Also, we present the relations of their solutions. Finally, through a simple numerical example, we illustrate our results. Particularly, in this example we find the optimal solution of the original problem by calculating the optimal solution of its associated semidefinite problem.

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Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIT) (NRF-2016R1A2B1006430). The authors would like to express their sincere thanks to anonymous referees for valuable suggestions and comments for the paper.

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  1. Department of Applied Mathematics, Pukyong National University, Busan, 48513, Republic of Korea

    Jae Hyoung Lee & Gue Myung Lee

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  1. Jae Hyoung Lee
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Correspondence to Gue Myung Lee.

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Lee, J.H., Lee, G.M. On minimizing difference of a SOS-convex polynomial and a support function over a SOS-concave matrix polynomial constraint. Math. Program. 169, 177–198 (2018). https://doi.org/10.1007/s10107-017-1210-z

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