Low nonconvex rank bilinear matrix inequalities: algorithms and applications

A branch and bound (BB) algorithm for solving a general class of bilinear matrix inequality (BMI) problem is proposed. First, linear matrix inequality (LMI) constraints are incorporated into BMI constraints in a special way to take advantage of useful informations on nonconvex terms. Then, the nonconvexity of BMI is centralized in coupling constraints so that when the latter are omitted, we get a relaxed LMI problem for computing lower bounds. As in our previous developments, the branching is performed in a reduced dimensional space of complicating variables. This makes the approach practical even with a large dimension of overall variables. Applications of the algorithm to several test problems of robust control are discussed