André Weng-Law, Paul-Emile Maingé, Fast inertial algorithms with a correction term for structured minimization problems, Vol. 2027 (2027), No. 16, pp. 1-29

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DOI: 10.23952/cot.2027.16

Received January 6, 2025; Accepted March 20, 2025; Published online October 22, 2026

Abstract. In this paper, we develop and study a new class of rapidly convergent proximal-like algorithms CRIPGA for solving structured non-smooth and convex minimization problems in a real Hilbert space framework. It can be regarded as a discrete counterpart of some recently discussed continuous Newton-like dynamical system with remarkable convergence properties. This new numerical scheme incorporates a correction term, besides the typical momentum term (arising in acceleration techniques introduced by Nesterov). As a special case of the parameters, we recover the very recently investigated method RIGPA (with no occurrence of correction term), for which interesting convergence properties were first established in the context of non-smooth minimization. A main specificity of both these latter processes is that they generate sequences that not only converge (towards equilibria) with some property of fast decaying subgradients (well-known as an uncommon efficient stability phenomenon), but also that minimize the function values at the rate $o(n^{-2})$ (as for the most famous related methods). However, our algorithm provides us some additional degree of freedom (through the correction term) that considerably influences its convergence properties. Numerical experiments are furthermore performed so as to illustrate the efficiency of CRIPGA compared with both classical and recently studied methods.

How to Cite this Article:
A. Weng-Law, P.E. Maingé, Fast inertial algorithms with a correction term for structured minimization problems, Commun. Optim. Theory 2027 (2027) 16.