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The appeals of quadratic majorization–minimization

Author

Listed:
  • Marc C. Robini

    (INSA Lyon)

  • Lihui Wang

    (Guizhou University)

  • Yuemin Zhu

    (INSA Lyon)

Abstract

Majorization–minimization (MM) is a versatile optimization technique that operates on surrogate functions satisfying tangency and domination conditions. Our focus is on differentiable optimization using inexact MM with quadratic surrogates, which amounts to approximately solving a sequence of symmetric positive definite systems. We begin by investigating the convergence properties of this process, from subconvergence to R-linear convergence, with emphasis on tame objectives. Then we provide a numerically stable implementation based on truncated conjugate gradient. Applications to multidimensional scaling and regularized inversion are discussed and illustrated through numerical experiments on graph layout and X-ray tomography. In the end, quadratic MM not only offers solid guarantees of convergence and stability, but is robust to the choice of its control parameters.

Suggested Citation

  • Marc C. Robini & Lihui Wang & Yuemin Zhu, 2024. "The appeals of quadratic majorization–minimization," Journal of Global Optimization, Springer, vol. 89(3), pages 509-558, July.
    • Handle: RePEc:spr:jglopt:v:89:y:2024:i:3:d:10.1007_s10898-023-01361-1
    DOI: 10.1007/s10898-023-01361-1
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    References listed on IDEAS

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    1. Groenen, Patrick J. F. & van de Velden, Michel, 2016. "Multidimensional Scaling by Majorization: A Review," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 73(i08).
    2. Jérôme Bolte & Edouard Pauwels, 2016. "Majorization-Minimization Procedures and Convergence of SQP Methods for Semi-Algebraic and Tame Programs," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 442-465, May.
    3. Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2014. "Variable Metric Forward–Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 107-132, July.
    4. Jan Leeuw, 1984. "Differentiability of Kruskal's stress at a local minimum," Psychometrika, Springer;The Psychometric Society, vol. 49(1), pages 111-113, March.
    5. Jan Leeuw, 1988. "Convergence of the majorization method for multidimensional scaling," Journal of Classification, Springer;The Classification Society, vol. 5(2), pages 163-180, September.
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