Global optimization methods for the bilinear matrix inequality problem

A number of important problems from system and control theory have been shown to be reducible to the bilinear matrix inequality (BMI) problem. In general, the BMI problem can be reformulated as a nonconvex programming problem, and therefore a global optimization method is required. Several global methods for the BMI problem and their modified versions have been proposed in the last 5 years. The purpose of this article is to provide an overview of the state of the art of global optimization methods for the BMI problem which include a primal relaxed-dual method, a branch and bound method, concave programming and Dual constrained programming.