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

Authors:
C. W. Scherer;C. W. J. Hol
Affiliations:
Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628, Delft, CD, The Netherlands;Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628, Delft, CD, The Netherlands
Venue:
Mathematical Programming: Series A and B
Year:
2006

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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.