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On the Approximation of Unbounded Convex Sets by Polyhedra

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  • Daniel Dörfler

    (Friedrich Schiller University Jena)

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.

Suggested Citation

  • Daniel Dörfler, 2022. "On the Approximation of Unbounded Convex Sets by Polyhedra," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 265-287, July.
    • Handle: RePEc:spr:joptap:v:194:y:2022:i:1:d:10.1007_s10957-022-02020-3
    DOI: 10.1007/s10957-022-02020-3
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    References listed on IDEAS

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    1. Andreas Löhne & Birgit Rudloff & Firdevs Ulus, 2014. "Primal and dual approximation algorithms for convex vector optimization problems," Journal of Global Optimization, Springer, vol. 60(4), pages 713-736, December.
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    5. Firdevs Ulus, 2018. "Tractability of convex vector optimization problems in the sense of polyhedral approximations," Journal of Global Optimization, Springer, vol. 72(4), pages 731-742, December.
    6. S. Ruzika & M. M. Wiecek, 2005. "Approximation Methods in Multiobjective Programming," Journal of Optimization Theory and Applications, Springer, vol. 126(3), pages 473-501, September.
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