The S-Procedure via dual cone calculus

Given a quadratic function h that satisfies a Slater condition, Yakubovich’s S-Procedure (or S-Lemma) gives a characterization of all other quadratic functions that are copositive with $h$ in a form that is amenable to numerical computations. In this paper we present a deep-rooted connection between the S-Procedure and the dual cone calculus formula $(K_{1} \cap K_{2})^{*} = K^{*}_{1} + K^{*}_{2}$, which holds for closed convex cones in $R^{2}$. To establish the link with the S-Procedure, we generalize the dual cone calculus formula to a situation where $K_{1}$ is nonclosed, nonconvex and nonconic but exhibits sufficient mathematical resemblance to a closed convex one. As a result, we obtain a new proof of the S-Lemma and an extension to Hilbert space kernels