A convex approach to a class of minimum norm problems

This paper considers the problem of determining the minimum euclidean distance of a point from a polynomial surface in R n. It is well known that this problem is in general non-convex. The main purpose of the paper is to investigate to what extent Linear Matrix Inequality (LMI) techniques can be exploited for solving this problem. The first result of the paper shows that a lower bound to the global minimum can be achieved via the solution of a one-parameter family of LMIs. Each LMI problem consists in the minimization of the maximum eigen value of a symmetric matrix. It is also pointed out that for some classes of problems the solution of a single LMI provides the lower bound. The second result concerns the tightness of the bound. It is shown that optimality of the lower bound can be readily checked via the solution of a system of linear equations. In addition, it is pointed out that lower bound tightness is strictly related to some properties concerning real homogeneous forms. Finally, an application example is developed throughout the paper to show the features of the approach.