The Mathematical Institute, University of Oxford, Eprints Archive

The S-Procedure via dual cone calculus

Hauser, Raphael (2013) The S-Procedure via dual cone calculus. Technical Report. Unspecified. (Submitted)

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

Item Type:Technical Report (Technical Report)
Subjects:O - Z > Operations research, mathematical programming
A - C > Calculus of variations and optimal control
Research Groups:Numerical Analysis Group
ID Code:1700
Deposited By: Lotti Ekert
Deposited On:14 May 2013 11:30
Last Modified:29 May 2015 19:23

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