Sequential parametric convex approximation algorithm for bilinear matrix inequality problem

The goal of this paper is to study algorithms for solving optimization problems subject to bilinear matrix inequalities (BMIs). This class of problems is known to be of great importance in engineering applications, for instance, control system designs. A main contribution is the development of a sequential convex optimization algorithm, where at each iteration step, a convex subproblem with linear matrix inequality (LMI) constraints is solved. The set of feasible points of the LMIs is a convex inner approximation of the set of feasible points of the BMI constraints around the current iteration point. Another contribution is the convergence proof of a subsequence of the iterations to a stationary point. Finally, an example of the static output-feedback controller design problem is given for comparative analysis.