The design centering problem as a D.C. programming problem

The following problem is studied: Given a compact setS inR ⁿ and a Minkowski functionalp(x), find the largest positive numberr for which there existsx S such that the set of ally R ⁿ satisfyingp(y–x) r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this design centering problem can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR ⁿ . In the case where, moreover,p(x) = (x ^T Ax)^1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.