On the Approximation of Unbounded Convex Sets by Polyhedra

Daniel Dörfler ()
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Daniel Dörfler: Friedrich Schiller University Jena

Journal of Optimization Theory and Applications, 2022, vol. 194, issue 1, No 11, 265-287

Abstract: Abstract This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the connections between existing results before introducing a new notion of polyhedral approximation called $$\left( \varepsilon , \delta \right) $$ ε , δ -approximation that integrates the unbounded case in a meaningful way. Some basic results about $$\left( \varepsilon , \delta \right) $$ ε , δ -approximations are proved for general convex sets. In the last section, an algorithm for the computation of $$\left( \varepsilon , \delta \right) $$ ε , δ -approximations of spectrahedra is presented. Correctness and finiteness of the algorithm are proved.

Keywords: Polyhedral approximation; Convex analysis; Unbounded sets; Algorithms; Spectrahedra; 52A27; 52B99; 90C22; 90C59 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-022-02020-3

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