DC Programming Approaches for BMI and QMI Feasibility Problems

We propose some new DC (difference of convex functions) programming approaches for solving the Bilinear Matrix Inequality (BMI) Feasibility Problems and the Quadratic Matrix Inequality (QMI) Feasibility Problems. They are both important NP-hard problems in the field of robust control and system theory. The inherent difficulty lies in the nonconvex set of feasible solutions. In this paper, we will firstly reformulate these problems as a DC program (minimization of a concave function over a convex set). Then efficient approaches based on the DC Algorithm (DCA) are proposed for the numerical solution. A semidefinite program (SDP) is required to be solved during each iteration of our algorithm. Moreover, a hybrid method combining DCA with an adaptive Branch and Bound is established for guaranteeing the feasibility of the BMI and QMI. A concept of partial solution of SDP via DCA is proposed to improve the convergence of our algorithm when handling more large-scale cases. Numerical simulations of the proposed approaches and comparison with PENBMI are also reported.