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Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs

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Abstract

We consider robust semi-definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description. On the basis of matrix sum-of-squares decompositions, we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart. With a novel matrix-version of Putinar's sum-of-squares representation for positive polynomials on compact semi-algebraic sets, we prove asymptotic exactness of the relaxation family under a suitable constraint qualification. If the uncertainty region is a compact polytope, we provide a new duality proof for the validity of Putinar's constraint qualification with an a priori degree bound on the polynomial certificates. Finally, we point out the consequences of our results for constructing relaxations based on the so-called full-block S-procedure, which allows to apply recently developed tests in order to computationally verify the exactness of possibly small-sized relaxations.

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Authors and Affiliations

  1. Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628, CD, Delft, The Netherlands

    C. W. Scherer & C. W. J. Hol

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  1. C. W. Scherer
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  2. C. W. J. Hol
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Supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs.

Sponsored by Philips CFT.

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Scherer, C., Hol, C. Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs. Math. Program. 107, 189–211 (2006). https://doi.org/10.1007/s10107-005-0684-2

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