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Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution

Author

Listed:
  • Hedy Attouch

    (IMAG, Univ. Montpellier, CNRS)

  • Szilárd Csaba László

    (Technical University of Cluj-Napoca)

Abstract

In a Hilbertian framework, for the minimization of a general convex differentiable function f, we introduce new inertial dynamics and algorithms that generate trajectories and iterates that converge fastly towards the minimizer of f with minimum norm. Our study is based on the non-autonomous version of the Polyak heavy ball method, which, at time t, is associated with the strongly convex function obtained by adding to f a Tikhonov regularization term with vanishing coefficient $$\varepsilon (t)$$ ε ( t ) . In this dynamic, the damping coefficient is proportional to the square root of the Tikhonov regularization parameter $$\varepsilon (t)$$ ε ( t ) . By adjusting the speed of convergence of $$\varepsilon (t)$$ ε ( t ) towards zero, we will obtain both rapid convergence towards the infimal value of f, and the strong convergence of the trajectories towards the element of minimum norm of the set of minimizers of f. In particular, we obtain an improved version of the dynamic of Su-Boyd-Candès for the accelerated gradient method of Nesterov. This study naturally leads to corresponding first-order algorithms obtained by temporal discretization. In the case of a proper lower semicontinuous and convex function f, we study the proximal algorithms in detail, and show that they benefit from similar properties.

Suggested Citation

  • Hedy Attouch & Szilárd Csaba László, 2024. "Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 99(3), pages 307-347, June.
    • Handle: RePEc:spr:mathme:v:99:y:2024:i:3:d:10.1007_s00186-024-00867-y
    DOI: 10.1007/s00186-024-00867-y
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    References listed on IDEAS

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    1. R. Cominetti, 1997. "Coupling the Proximal Point Algorithm with Approximation Methods," Journal of Optimization Theory and Applications, Springer, vol. 95(3), pages 581-600, December.
    2. A. Cabot, 2004. "Inertial Gradient-Like Dynamical System Controlled by a Stabilizing Term," Journal of Optimization Theory and Applications, Springer, vol. 120(2), pages 275-303, February.
    3. Baillon, J.B. & Cominetti, R., 2001. "A Convergence Result for Non-Autonomous Subgradient Evolution Equations and Its Application to the Steepest Descent Exponential Penality Trajectory in Linear Programming," Papiers d'Economie Mathématique et Applications 2001.33, Université Panthéon-Sorbonne (Paris 1).
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