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Self-concordance and matrix monotonicity with applications to quantum entanglement problems

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  • Faybusovich, Leonid
  • Zhou, Cunlu

Abstract

Let V be a Euclidean Jordan algebra and Ω be a cone of invertible squares in V. Suppose that g:R+→R is a matrix monotone function on the positive semiaxis which naturally induces a function g˜:Ω→V. We show that −g˜ is compatible (in the sense of Nesterov–Nemirovski) with the standard self-concordant barrier B(x)=−lndet(x) on Ω. As a consequence, we show that for any c ∈ Ω, the functions of the form −tr(cg˜(x))+B(x) are self-concordant on Ω. In particular, the function x↦−tr(clnx) is a self-concordant barrier function on Ω. Using these results, we apply a long-step path-following algorithm developed in [L. Faybusovich and C. Zhou Long-step path-following algorithm for solving symmetric programming problems with nonlinear objective functions. Comput Optim Appl, 72(3):769-795, 2019] to a number of important optimization problems arising in quantum information theory. Results of numerical experiments and comparisons with existing methods are presented.

Suggested Citation

  • Faybusovich, Leonid & Zhou, Cunlu, 2020. "Self-concordance and matrix monotonicity with applications to quantum entanglement problems," Applied Mathematics and Computation, Elsevier, vol. 375(C).
    • Handle: RePEc:eee:apmaco:v:375:y:2020:i:c:s0096300320300400
    DOI: 10.1016/j.amc.2020.125071
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    References listed on IDEAS

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    1. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, June.
    2. Leonid Faybusovich & Cunlu Zhou, 2019. "Long-step path-following algorithm for solving symmetric programming problems with nonlinear objective functions," Computational Optimization and Applications, Springer, vol. 72(3), pages 769-795, April.
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