The projective method for solving linear matrix inequalities

In many control problems, the design constraints have natural formulations in terms of linear matrix inequalities (LMI). When no analytical solution is available, such problems can be attacked by solving the LMIs via convex optimization techniques. This paper describes the polynomial-time projective algorithm for the numerical solution of LMIs. Simple geometrical arguments are used to clarify the strategy and convergence mechanism of the projective method. A complexity analysis is provided, and applications to two generic LMI problems are discussed.