A Region-Dividing Technique for Constructing the Sum-of-Squares Approximations to Robust Semidefinite Programs

In this technical note, we present a novel approach to robust semidefinite programs, of which coefficient matrices depend polynomially on uncertain parameters. The approach is based on approximation with the sum-of-squares polynomials, but, in contrast to the conventional sum-of-squares approach, the quality of approximation is improved by dividing the parameter region into several subregions. The optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer. An advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division. A numerical example on polynomial optimization is presented to show usefulness of the present approach.