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Tight bounds on Lyapunov rank

Optimization Letters volume 16, pages 723–728 (2022)Cite this article

Abstract

The Lyapunov rank of a cone is the number of independent equations obtainable from an analogue of the complementary slackness condition in cone programming problems, and more equations are generally thought to be better. Bounding the Lyapunov rank of a proper cone in \( \mathbb {R}^{n}\) from above has been an open problem. Gowda and Tao gave an upper bound of \(n^{2} - n\) that was later improved by Orlitzky and Gowda to \( \left( {n-1}\right) ^{2}\). We settle the matter and show that the Lyapunov rank of \( \left( {n^{2} - n}\right) /2 + 1\) corresponding to the Lorentz cone is maximal.

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  1. Department of Mathematics, Towson University, 7800 York Road, Towson, MD, 21252, USA

    Michael Orlitzky

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  1. Michael Orlitzky
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Correspondence to Michael Orlitzky.

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Orlitzky, M. Tight bounds on Lyapunov rank . Optim Lett 16, 723–728 (2022). https://doi.org/10.1007/s11590-021-01750-z

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  • DOI: https://doi.org/10.1007/s11590-021-01750-z

Keywords

  • Lyapunov rank
  • Bilinearity rank
  • Lorentz cone
  • Proper cone