On convexification of some minimum distance problems

This paper considers the problem of determining the minimum euclidean distance of a point from a polynomial surface in Rn. 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 Linear Matrix Inequalities (LMIs). It is also pointed out that for some classes of problems the solution of a single LMI problem provides the lower bound. The second result concerns the tightness of the bound. It is shown that optimality of the lower bound can be easily checked via the solution of a system of linear equations. Two application examples are finally presented to show potentialities of the approach.