Reduction of the Number of Constraints in the Matrix-Dilation Approach to Robust Semidefinite Programming

In order to solve a robust semidefinite programming problem, which is a semidefinite programming problem containing uncertain parameters, an approximate problem with a small number of constraints is proposed. The proposed approximate problem is constructed by dilation of the matrices in the constraints. The minimum value of the approximate problem converges to that of the original problem as the parameter region is divided finer. Although, with the previously proposed approximate problem, the number of constraints is of exponential order in the parameter dimension, it is of linear order with the new approximate problem. Just as the previous approximate problem, the quality of approximation is expressed in terms of the resolution of the division. This result is obtained by application of the result of Ben-Tal-Nemirovski (2002). In the last section of the paper, a useful property is derived on polynomial optimization, which can be reduced to robust semidefinite programming