Exploiting sparsity in the sum-of-squares approximations to robust semidefinite programs

Abstract

This paper aims to improve computational complexity in the sum-of-squares approximations to robust semidefinite programs whose constraints depend polynomially on uncertain parameters. By exploiting sparsity, the proposed approach constructs sum-of-squares polynomials with smaller number of monomial elements, and hence gives approximate problems with smaller sizes. The sparse structure is extracted by a special graph pattern. The quality of the approximation is improved by dividing the parameter region, and can be expressed in terms of the resolution of the division. This expression shows that the proposed approach is asymptotically exact in the sense that, the quality can be arbitrarily improved by increasing the resolution of the division.

Polynomial sum of squares in fluid dynamics: a review with a look ahead

“…A variety of well-supported software codes for solving such problems are freely available, both for SOS problems in particular [6] and for SDP problems in general [7,8]. More recently, research has focused on the efficient solution of SDP problems arising from SOS problems, with particular emphasis on robust optimization [9,10], exploitation of structure and sparsity [11][12][13] and decomposition techniques [14] in large problems. As a result of these advances, the SOS approach has found extensive applications in stability analysis, control theory and many other fields, including applications in aeronautics [15,16].…”