arXiv subjects

César López-Pastor

Publications and source records attributed to César López-Pastor.

6 recordsLinked to original sources

arXiv2026

Quasi-pseudometric modular spaces as $\mathscr{Q}$-categories

We prove that the category of quasi-pseudometric modular spaces whose morphisms are the nonexpansive mappings is isomorphic to a quantale enriched category. To achieve this, we construct an appropriate quantale of isotone functions. We also show that, by means of this isomorphism, the topology associated with a quasi-pseudometric modular coincides with that generated by its corresponding quantale enriched category. Furthermore, we demonstrate that the class of quasi-pseudometrizable topological spaces coincides with the topological spaces whose topology is induced by a quasi-pseudometric modular.

math.CT↗

arXiv2026

On the optimal relaxation parameter of graph-based splitting methods for subspaces

In this paper, we investigate the behavior of the family of graph-based splitting algorithms specialized to the problem of finding a point in the intersection of linear subspaces. The algorithms in this family, which encompasses several classical methods such as the Douglas-Rachford algorithm, are defined by a connected graph and a subgraph. Our main result establishes that when the graph and subgraph coincide, the optimal relaxation parameter is exactly $1$, thereby extending known results for the Douglas-Rachford algorithm to a much broader class of methods. Our analysis hinges on some properties of iso-averaged linear operators, which are defined as the average of an isometry and the identity, and are characterized by a specific symmetry of the norm of their relaxation.

math.OC↗

arXiv2026

Graph splitting methods: Fixed points and strong convergence for linear subspaces

In this paper, we develop a general analysis for the fixed points of the operators defining the graph splitting methods from [SIAM J. Optim., 34 (2024), pp. 1569-1594] by Bredies, Chenchene and Naldi. We particularize it to the case where the maximally monotone operators are normal cones of closed linear subspaces and provide an explicit formula for the limit points of the graph splitting schemes. We exemplify these results on some particular algorithms, unifying in this way some results previously derived as well as obtaining new ones.

math.OC↗

arXiv2026

A comparative numerical study of graph-based splitting algorithms for linear subspaces

In this note, we test the performance of six algorithms from the family of graph-based splitting methods [SIAM J. Optim., 34 (2024), pp. 1569-1594] specialized to normal cones of linear subspaces. To do this, we first implement some numerical experiments to determine the best relaxation parameter for each algorithm. Then, we compare the number of iterations each algorithm requires to reach a given stopping criterion, using the previously identified best relaxation parameter. The numerical results allow us to identify some relevant patterns and provide numerical evidence that may guide further theoretical analysis.

math.NA↗

arXiv2025

Projection onto the parabola

In this note, we provide explicit expressions for the projections onto the graph of a quadratic polynomial. The projections are obtained by examining the critical points of the associated quartic polynomial, that is, the roots of the cubic polynomial defining its derivative. We also focus on the case where the point we project lies on the vertical line defined by the parabola. Lastly, an explicit formula for the projection onto a higher dimensional parabola is derived.

math.GM↗

arXiv2024

Forward-backward algorithms devised by graphs

In this work, we present a methodology for devising forward-backward methods for finding zeros in the sum of a finite number of maximally monotone operators. We extend the framework and techniques from [SIAM J. Optim., 34 (2024), pp. 1569-1594] to cover the case involving a finite number of cocoercive operators, which should be directly evaluated instead of computing their resolvent. The algorithms are induced by three graphs that determine how the algorithm variables interact with each other and how they are combined to compute each resolvent. The hypotheses on these graphs ensure that the algorithms obtained have minimal lifting and are frugal, meaning that the ambient space of the underlying fixed point operator has minimal dimension and that each resolvent and each cocoercive operator is evaluated only once per iteration. This framework not only allows to recover some known methods, but also to generate new ones, as the forward-backward algorithm induced by a complete graph. We conclude with a numerical experiment showing how the choice of graphs influences the performance of the algorithms.

math.OC↗