arXiv subjects

Javier I. Madariaga

Publications and source records attributed to Javier I. Madariaga.

6 recordsLinked to original sources

arXiv2026

Asymptotic Analysis of an Abstract Stochastic Scheme for Solving Monotone Inclusions

We propose an abstract stochastic scheme for solving a broad class of monotone operator inclusion problems in Hilbert spaces. It is based on a reduced inclusion model with two operators. At each iteration of the abstract algorithm, stochastic approximations of points in the graphs of these operators are used to construct the update. We establish the almost sure and $L^2$ convergence of the iterates to a solution of the underlying inclusion problem. A key feature of the proposed framework is its flexibility as it can accommodate stochasticity at several levels: the approximation of operators, the selection of coordinates and operators in block-iterative implementations, and the choice of relaxation parameters. This makes it well suited for the asymptotic analysis of a wide range of stochastic monotone operator splitting methods. The broad scope of the proposed framework is illustrated by applications to various splitting methods. Thus, we derive new results on the almost sure and $L^2$ convergence of stochastic versions of the proximal point algorithm, as well as of randomized block-iterative Kuhn--Tucker and saddle projective splitting methods for solving systems of coupled inclusions involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators combined via various monotonicity-preserving operations. Specific applications are discussed.

math.OC↗

arXiv2026

Almost Supermartingale Extensions of Olivier's Theorem

Olivier's 1827 theorem provides a rate of convergence to zero of the general term of a decreasing summable sequence of positive reals. We derive stochastic extensions of this result in the context of almost supermartingales. The results are applied to the analysis of stochastic iterative processes.

math.PR↗

arXiv2026

Convergence of the Iterates of the Stochastic Proximal Gradient Method

We propose a novel study of the stochastic proximal gradient method for minimizing the sum of two convex functions, one of which is smooth. Under suitable assumptions and without requiring any boundedness or control of the variance of the random variables, we derive the almost sure convergence and the convergence in the mean of the iterates to a solution of the minimization problem. The results are applied to classification and convex feasibility problems.

math.OC↗

arXiv2026

An Abstract Stochastic Haugazeau Method for Best Approximation

The Haugazeau method was originally designed to compute the best approximation from an intersection of closed convex sets in Hilbert spaces using the projection operators onto the individual sets iteratively. We propose an abstract stochastic version of it to compute the best approximation from a closed convex set by successive projections onto randomly generated stochastic outer approximations of that set. Strong convergence in the mean square and the almost sure modes is derived under general hypotheses on the outer approximations. The results are applied to the development of stochastic algorithms to construct the best approximation from an arbitrary intersection of fixed point sets by random activation of blocks of operators. A numerical application to the computation of Chebyshev centers is provided.

math.OC↗

arXiv2026

A Geometric Framework for Stochastic Iterations

This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which existing solution methods can be recast and improved, and new ones can be designed. Almost sure weak, strong, and linear convergence results are established in particular for stochastic fixed point iterations, the stochastic gradient descent method, and stochastic extrapolated parallel algorithms for feasibility problems. In these areas, the proposed algorithms exceed the features and convergence guarantees of the state of the art. Numerical applications to signal and image recovery are provided.

math.OC↗

arXiv2025

Almost-Surely Convergent Randomly Activated Monotone Operator Splitting Methods

We propose stochastic splitting algorithms for solving large-scale composite inclusion problems involving monotone and linear operators. They activate at each iteration blocks of randomly selected resolvents of monotone operators and, unlike existing methods, achieve almost sure convergence of the iterates to a solution without any regularity assumptions or knowledge of the norms of the linear operators. Applications to image recovery and machine learning are provided.

math.OC↗