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

• Published in: Mathematical programming Vol. 107; no. 1-2; pp. 189 - 211
• Main Authors: Scherer, C. W., Hol, C. W. J.
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01-06-2006
• Subjects: Digital Object Identifier; Inequality; Linear matrix inequalities; Mathematical analysis; Mathematical programming; Matrix; Parameter uncertainty; Polynomial matrices; Polytopes; Robust control; Robustness; Studies; Upper bounds
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-005-0684-2

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. [PUBLICATION ABSTRACT]