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

This paper aims to improve computational complexity in the sum-of-squares approximations to robust semi-definite 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.