The S-Procedure via Dual Cone Calculus

TL;DR

This paper explores a novel connection between the S-Procedure and dual cone calculus, extending the classical S-Lemma to more general settings including nonclosed, nonconvex cones and Hilbert space kernels.

Contribution

It introduces a new link between the S-Procedure and dual cone calculus, generalizes the dual cone formula, and provides an extended proof of the S-Lemma applicable to broader contexts.

Findings

Established a connection between the S-Procedure and dual cone calculus.

Generalized the dual cone formula to nonclosed, nonconvex sets.

Extended the S-Lemma to Hilbert space kernels.

Abstract

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 cone. As a result, we obtain a new proof of the S-Lemma and an extension to Hilbert space kernels.