Nonsmooth methods for large bilinear matrix inequalities: Applications to feedback control

We present a method to solve large optimization programs with bilinear matrix inequality (BMI) constraints. Such programs arise in control applications such as system analysis, controller synthesis or filter design. Our specific point of view here is to cast BMI problems as nonconvex eigenvalue optimization programs and to use nonsmooth optimization methods suited for eigenvalue optimization to compute locally optimal solutions. Our method is based on the !-subgradient prototype, suitably adapted to include non-convex problems. In each tangent step, a small size semidefinite program (SDP) is solved to compute a search direction. Our method is tested on several large scale benchmark problems in output feedback controller design.