Polyhedral Approximation of Spectrahedral Shadows via Homogenization

Daniel Dörfler () and Andreas Löhne ()
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Daniel Dörfler: Friedrich Schiller University Jena
Andreas Löhne: Friedrich Schiller University Jena

Journal of Optimization Theory and Applications, 2024, vol. 200, issue 2, No 17, 874-890

Abstract: Abstract This article is concerned with the problem of approximating a not necessarily bounded spectrahedral shadow, a certain convex set, by polyhedra. By identifying the set with its homogenization, the problem is reduced to the approximation of a closed convex cone. We introduce the notion of homogeneous $$\delta $$ δ -approximation of a convex set and show that it defines a meaningful concept in the sense that approximations converge to the original set if the approximation error $$\delta $$ δ diminishes. Moreover, we show that a homogeneous $$\delta $$ δ -approximation of the polar of a convex set is immediately available from an approximation of the set itself under mild conditions. Finally, we present an algorithm for the computation of homogeneous $$\delta $$ δ -approximations of spectrahedral shadows and demonstrate it on examples.

Keywords: Polyhedral approximation; Unbounded convex sets; Spectrahedral shadows; Homogenization; 52A27; 90C59 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-023-02363-5

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