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Convergence of the Exponentiated Gradient Method with Armijo Line Search

Author

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  • Yen-Huan Li

    (École Polytechnique Fédérale de Lausanne
    National Taiwan University)

  • Volkan Cevher

    (École Polytechnique Fédérale de Lausanne)

Abstract

Consider the problem of minimizing a convex differentiable function on the probability simplex, spectrahedron, or set of quantum density matrices. We prove that the exponentiated gradient method with Armijo line search always converges to the optimum, if the sequence of the iterates possesses a strictly positive limit point (element-wise for the vector case, and with respect to the Löwner partial ordering for the matrix case). To the best of our knowledge, this is the first convergence result for a mirror descent-type method that only requires differentiability. The proof exploits self-concordant likeness of the log-partition function, which is of independent interest.

Suggested Citation

  • Yen-Huan Li & Volkan Cevher, 2019. "Convergence of the Exponentiated Gradient Method with Armijo Line Search," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 588-607, May.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:2:d:10.1007_s10957-018-1428-9
    DOI: 10.1007/s10957-018-1428-9
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    References listed on IDEAS

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    1. Charles Byrne & Yair Censor, 2001. "Proximity Function Minimization Using Multiple Bregman Projections, with Applications to Split Feasibility and Kullback–Leibler Distance Minimization," Annals of Operations Research, Springer, vol. 105(1), pages 77-98, July.
    2. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
    3. A. Auslender & M. Teboulle, 2004. "Interior Gradient and Epsilon-Subgradient Descent Methods for Constrained Convex Minimization," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 1-26, February.
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    Cited by:

    1. Andrzej Cichocki & Sergio Cruces & Auxiliadora Sarmiento & Toshihisa Tanaka, 2024. "Generalized Exponentiated Gradient Algorithms and Their Application to On-Line Portfolio Selection," Papers 2406.00655, arXiv.org.

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