Global optimization: a model problem

where PQ denotes the polynomial multiplication (convolution). For given n this is an optimization problem in 2n unknowns with polynomials P,Q of degree n− 1. Denote the global minimum by μn. We have a very efficient algorithm to compute an upper bound for μn, and we strongly believe that our approximations are very close to the true minimum. The challenge is to compute rigorous lower bounds for μn. The best known lower and upper bounds for μn are displayed in Table 1. As has been mentioned, we have reasons to believe that the displayed upper bounds are very close to the global minimum. By mathematical means we can show (see below) that global minimizers P, Q must have all roots on the unit circle. More precisely, for arbitrary given normed P a (normed) polynomial Q minimizing ||PQ|| has all its roots on the unit circle. ∗Institute for Reliable Computing, Hamburg University of Technology, Schwarzenbergstraße 95, Hamburg 21071, Germany, and Visiting Professor at Waseda University, Faculty of Science and Engineering, 3–4–1 Okubo, Shinjuku-ku, Tokyo 169–8555, Japan ([email protected]).