A region-dividing approach to robust semidefinite programming and its error bound

Abstract

A new asymptotically exact approach is presented for robust semidefinite programming, where coefficient matrices polynomially depend on uncertain parameters. Since a robust semidefinite programming problem is difficult to solve directly, an approximate problem is constructed based on a division of the parameter region. 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 is available before solving the approximate problem. This bound shows how the approximation error depends on the resolution of the division. Furthermore, it leads to construction of an efficient division that attains small approximation error with low computational complexity. Numerical examples show efficacy of the present approach. In particular, an exact optimal value is often found with a division of finite resolution.