A primal-dual potential reduction method for integral quadratic constraints

We discuss how to implement an efficient interior-point algorithm for semi-definite programs that result from integral quadratic constraints. The algorithm is a primal-dual potential reduction method, and the computational effort is dominated by a least-squares system that has to be solved in each iteration. The key to an efficient implementation is to utilize iterative methods and the specific structure of integral quadratic constraints. The algorithm has been implemented in Matlab. To give a rough idea of the efficiencies obtained, it is possible to solve problems resulting in a linear matrix inequality of dimension 130 × 130 with approximately 5000 variables in about 5 minutes on a lap-top. Problems with approximately 20000 variable and a linear matrix inequality of dimension 230 × 230 are solved in about 45 minutes. It is not assumed that the system matrix has no eigenvalues on the imaginary axis, nor is it assumed that it is Hurwitz