Minimizing trigonometric matrix polynomials over semi-algebraic sets

This paper addresses the problem of minimizing the minimum eigenvalue of a trigonometric matrix polynomial. The contribution is to show that, by exploiting Putinar’s Positivstellensatz and introducing suitable transformations, it is possible to derive a nonconservative approach based on semidefinite programming (SDP) whose computational burden can be significantly smaller than that of an existing method recently published. Other advantages of the proposed approach include the possibility of taking into account the presence of constraints in the form of semi-algebraic sets and establishing tightness of a found lower bound.