On the S-Lemma for Univariate Polynomials

The so-called S-lemma has played an important role in optimization, both in theory and in applications. The significance of S-lemma is especially pronounced in control theory, robust optimization, and non-convex quadratic optimization. Hitherto, S-lemma is however established only in the domain of quadratic functions. In this paper we shall extend the notion of S-lemma to the class of univariate polynomials of any degree. The applications of the new S-lemma will be discussed as well.