arXiv subjects

Juan José Maulén

Publications and source records attributed to Juan José Maulén.

6 recordsLinked to original sources

arXiv2026

Preconditioned primal-dual algorithms for saddle point problems: non-ergodic convergence rates

We study a family of preconditioned primal dual algorithms for convex-concave saddle point problems by the dynamics introduced in \cite{apidopoulos2026preconditioned}. The proposed framework exploits the possible smooth + nonsmooth structure of the saddle point formulation. It includes, but is not limited to, linearly constrained convex optimization problems. The proposed antisymmetric preconditioners allow us to establish non ergodic convergence rates, accounting for possible computational errors in the implementation of the method. Finally, we present numerical experiments to indicate our well performed preconditioned primal dual algorithms.

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arXiv2026

Towards faster first order methods: A continuous-time model to interpolate between speed and function value restart

We introduce a new restarting scheme for a continuous inertial dynamics with Hessian driven-damping, and establish a linear convergence rate for the function values along the restarted trajectories. The proposed routine is implemented without knowing the strong convexity parameter, and is a generalization of existing speed restart schemes. It interpolates between speed and function value restarts, considerably delaying the restarting time, while preserving convergence and function value decrease. Numerical experiments show an improvement in the convergence rates for both continuous-time dynamical systems, and the associated accelerated first-order algorithms derived via time discretization.

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arXiv2026

Penalty-Based Smoothing of Convex Nonsmooth Supremum Functions with Accelerated Inertial Dynamics

We propose a penalty-based smoothing framework for convex nonsmooth functions with a supremum structure. The regularization yields a differentiable surrogate with controlled approximation error, a single-valued dual maximizer, and explicit gradient formulas. We then study an accelerated inertial dynamic with vanishing damping driven by a time-dependent regularized function whose parameter decreases to zero. Under mild integrability and boundedness conditions on the regularization schedule, we establish an accelerated $\mathcal{O}(t^{-2})$ decay estimate for the regularized residual and, in the regime $α>3$, a sharper $o(t^{-2})$ decay together with weak convergence of trajectories to a minimizer of the original nonsmooth problem via an Opial-type argument. Applications to multiobjective optimization (through Chebyshev/max scalarization) and to distributionally robust optimization (via entropic regularization over ambiguity sets) illustrate the scope of the framework.

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arXiv2025

Relaxed and inertial nonlinear Forward-Backward algorithm

The Nonlinear Forward-Backward (NFB) algorithm, also known as warped resolvent iterations, is a splitting method for finding zeros of sums of monotone operators. In particular cases, NFB reduces to well-known algorithms such as Forward-Backward, Forward-Backward-Forward, Chambolle--Pock, and Condat--Vũ. Therefore, NFB can be used to solve monotone inclusions involving sums of maximally monotone, cocoercive, monotone and Lipschitz operators as well as linear compositions terms. In this article, we study the weak and strong (linear) convergence of NFB with inertial and relaxation steps. Our results recover known convergence guarantees for the aforementioned methods when extended with inertial and relaxation terms. Additionally, we establish the convergence of inertial and relaxed variants of the Forward-Backward-Half-Forward and Forward-Primal--Dual-Half-Forward algorithms, which, to the best of our knowledge, are new contributions. We consider both nondecreasing and decreasing sequences of inertial parameters, the latter being a novel approach in the context of inertial algorithms. To evaluate the performance of these strategies, we present numerical experiments on optimization problems with affine constraints and on image restoration tasks. Our results show that decreasing inertial sequences can accelerate the numerical convergence of the algorithms.

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arXiv2024

A Speed Restart Scheme for a Dynamical System with Hessian-Driven Damping and Three Constant Coefficients

In this paper, we study a speed restart scheme for an inertial system with Hessian-driven damping. We establish a linear convergence rate for the function values along the restarted trajectories without assuming the strong convexity of the objective function. Our numerical experiments show improvements in the convergence rates, both for the continuous-time dynamics, and when applied to inertial algorithms as a heuristic

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arXiv2023

Inertial Krasnoselskii-Mann Iterations

We establish the weak convergence of inertial Krasnoselskii-Mann iterations towards a common fixed point of a family of quasi-nonexpansive operators, along with estimates for the non-asymptotic rate at which the residuals vanish. Strong and linear convergence are obtained in the quasi-contractive setting. In both cases, we highlight the relationship with the non-inertial case, and show that passing from one regime to the other is a continuous process in terms of the hypotheses on the parameters. Numerical illustrations are provided for an inertial primal-dual method and an inertial three-operator splitting algorithm, whose performance is superior to that of their non-inertial counterparts.

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