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Convex quartic problems: homogenized gradient method and preconditioning

Author

Listed:
  • Dragomir, Radu-Alexandru

    (EPFL)

  • Nesterov, Yurii

    (Université catholique de Louvain, LIDAM/CORE, Belgium)

Abstract

We consider a convex minimization problem for which the objective is the sum of a homogeneous polynomial of degree four and a linear term. Such task arises as a subproblem in algorithms for quadratic inverse problems with a difference-of-convex structure. We design a first-order method called Homogenized Gradient, along with an accelerated version, which enjoy fast convergence rates of respectively O(κ2/K2) and O(κ2/K4) in relative accuracy, where K is the iteration counter. The constant κ is the quartic condition number of the problem. Then, we show that for a certain class of problems, it is possible to compute a preconditioner n, where n is the problem dimension. To establish this, we study the more general problem of finding the best quadratic approximation of an lp norm composed with a quadratic map. Our construction involves a generalization of the so-called Lewis weights.

Suggested Citation

  • Dragomir, Radu-Alexandru & Nesterov, Yurii, 2024. "Convex quartic problems: homogenized gradient method and preconditioning," LIDAM Discussion Papers CORE 2024026, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2024026
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    References listed on IDEAS

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    1. Filip Hanzely & Peter Richtárik & Lin Xiao, 2021. "Accelerated Bregman proximal gradient methods for relatively smooth convex optimization," Computational Optimization and Applications, Springer, vol. 79(2), pages 405-440, June.
    2. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    3. Leonid G. Khachiyan, 1996. "Rounding of Polytopes in the Real Number Model of Computation," Mathematics of Operations Research, INFORMS, vol. 21(2), pages 307-320, May.
    4. ANSTREICHER, Kurt M., 1999. "Ellipsoidal approximations of convex sets based on the volumetric barrier," LIDAM Reprints CORE 1393, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. 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.
    6. K. M. Anstreicher, 1999. "Ellipsoidal Approximations of Convex Sets Based on the Volumetric Barrier," Mathematics of Operations Research, INFORMS, vol. 24(1), pages 193-203, February.
    7. Da Kuang & Sangwoon Yun & Haesun Park, 2015. "SymNMF: nonnegative low-rank approximation of a similarity matrix for graph clustering," Journal of Global Optimization, Springer, vol. 62(3), pages 545-574, July.
    8. NESTEROV Yurii,, 2020. "Superfast second-order methods for unconstrained convex optimization," LIDAM Discussion Papers CORE 2020007, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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