Matrix cones, complementarity problems, and the bilinear matrix inequality

Discusses an approach for solving the bilinear matrix inequality (BMI) based on its connections with certain problems defined over matrix cones. These problems are, among others, the cone generalization of the linear programming (LP) and the linear complementarity problem (LCP) (referred to as the Cone-LP and the Cone-LCP, respectively). Specifically, the authors show that solving a given BMI is equivalent to examining the solution set of a suitably constructed Cone-LP or Cone-LCP. This approach facilitates understanding of the geometry of the BMI and opens up new avenues for the development of the computational procedures for its solution.