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Accelerated forward–backward algorithms for structured monotone inclusions

Author

Listed:
  • Paul-Emile Maingé

    (F.W.I., MEMIAD, Université des Antilles)

  • André Weng-Law

    (F.W.I., MEMIAD, Université des Antilles)

Abstract

In this paper, we develop rapidly convergent forward–backward algorithms for computing zeroes of the sum of two maximally monotone operators. A modification of the classical forward–backward method is considered, by incorporating an inertial term (closed to the acceleration techniques introduced by Nesterov), a constant relaxation factor and a correction term, along with a preconditioning process. In a Hilbert space setting, we prove the weak convergence to equilibria of the iterates $$(x_n)$$ ( x n ) , with worst-case rates of $$ o(n^{-1})$$ o ( n - 1 ) in terms of both the discrete velocity and the fixed point residual, instead of the rates of $$\mathcal {O}(n^{-1/2})$$ O ( n - 1 / 2 ) classically established for related algorithms. Our procedure can be also adapted to more general monotone inclusions. In particular, we propose a fast primal-dual algorithmic solution to some class of convex-concave saddle point problems. In addition, we provide a well-adapted framework for solving this class of problems by means of standard proximal-like algorithms dedicated to structured monotone inclusions. Numerical experiments are also performed so as to enlighten the efficiency of the proposed strategy.

Suggested Citation

  • Paul-Emile Maingé & André Weng-Law, 2024. "Accelerated forward–backward algorithms for structured monotone inclusions," Computational Optimization and Applications, Springer, vol. 88(1), pages 167-215, May.
  • Handle: RePEc:spr:coopap:v:88:y:2024:i:1:d:10.1007_s10589-023-00547-3
    DOI: 10.1007/s10589-023-00547-3
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Laurent Condat, 2013. "A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 460-479, August.
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