Local structure and algorithms in nonsmooth optimization

Given a real δ, find stable real polynomials p and q such that the polynomial r(s) = (s − 2δs + 1)p(s) + (s − 1)q(s) is also stable. (We call a polynomial p stable if its abscissa α(p) = max{Re s : p(s) = 0} is nonpositive.) Clearly the problem is unsolvable if δ = 1, since then r(1) = 0; more delicate results (summarized in [7]) show it remains unsolvable for δ < 1 close to 1. Blondel offered a prize of 1kg of Belgian chocolate for the case δ = 0.9, a problem solved via randomized search in [7]. To illustrate the theme of this talk, we first outline (based on joint work with D. Henrion) a more systematic, optimization approach to the chocolate problem. We fix the degrees of the polynomials p and q (say 3, for example), without loss of generality suppose p is monic, and consider the resulting problem