Convex relaxations for quadratic distance problems

Convex relaxations of nonconvex problems are a powerful tool for the analysis and design of control systems. An important family of nonconvex problems that are relevant to the control field is that of quadratic distance problems. In this paper, several convex relaxations are presented for quadratic distance problems which are based on the sum-of squares representation of positive polynomials. Relationships among the considered relaxations are discussed and numerical comparisons are presented, in order to highlight their degree of conservatism.