A Cutting Plane Method for Solving Convex Optimization Problems over the Cone of Nonnegative Polynomials

Many practical problems can be formulated as convex optimization problems over the cone of nonnegative univariate polynomials. We use a cutting plane method for solving this type of optimization problems in primal form. Therefore, we must be able to verify whether a polynomial is nonnegative, i.e. if it does not have real roots or all real roots are multiple of even order. In this paper an efficient method is derived to determine a scalar value for which the polynomial is negative and in the case that such a value exists a feasible cut is constructed. Our method is based on Sturm theorem, which allows to determine the number of distinct roots of a polynomial on a given interval, in combination with the bisection method. For numerical stability we construct the associated Sturm sequence using Chebyshev basis, and thus we can work with high degree polynomials, up to hundreds. Numerical results show the efficiency of our new approach. Key–Words: convex optimization problems over the cone of nonnegative polynomials, cutting plane method, Chebyshev polynomials, Sturm sequence, feasible cut.