Journal of Nonlinear and Variational Analysis

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DOI: 10.23952/jnva.10.2026.5.06

Volume 10, Issue 5, 1 October 2026, Pages 965-989

 

Abstract. The primary objective of this paper is to achieve strong convergence toward a zero of a maximally monotone operator in a Hilbert space. To this end, we first propose a continuous dynamic system that builds upon the framework presented by Boţ and Nguyen (2025), while remaining distinct from it. Specifically, we apply a positive linear operator to the velocity term ẋ(t). This modification enables the adaptation of a suitable proximal algorithm when applying the proposed system to the context of the composite minimization problem f(x)+g(Ax), where f and g are convex functions and A is a linear operator, and then to convex minimization problems under linear constraints, via a primal-dual algorithm by Chambolle and Pock (2011) for saddle points of the associated Lagrangian. The continuous model is analyzed for a single-valued operator M, allowing us to illustrate -through an appropriate discretization- the corresponding proximal point algorithm for a set-valued operator and to provide a consistent proof for the suitable convergence rates. The algorithmic contribution of this work is particularly significant, as it establishes strong convergence to the minimum norm of zeros of M without requiring additional conditions on its maximal monotonicity, a result that remains absent from more recent literature. We then provide strong convergence to the minimum norm solution, and also the same rate of convergence for values and constraints for the composite minimization problem and convex minimization under linear constraints.

 

How to Cite this Article:
F. Battahi, Z. Chbani, H. Riahi, Preconditioned Tikhonov regularised monotone dynamical systems and applications to primal-dual algorithm in convex optimization, J. Nonlinear Var. Anal. 10 (2026), 965-989.