arXiv subjects

Genaro López-Acedo

Publications and source records attributed to Genaro López-Acedo.

8 recordsLinked to original sources

arXiv2026

Stochastic and incremental subgradient methods for convex optimization on Hadamard spaces

As a foundation for optimization, convexity is useful beyond the classical settings of Euclidean and Hilbert space. The broader arena of nonpositively curved metric spaces, which includes manifolds like hyperbolic space, as well as metric trees and more general CAT(0) cubical complexes, supports primal tools like proximal operations for geodesically convex functions. However, the lack of linear structure in such spaces complicates dual constructions like subgradients. To address this hurdle, we introduce a new type of subgradient for functions on Hadamard spaces, based on Busemann functions. Our notion supports generalizations of classical stochastic and incremental subgradient methods, with guaranteed complexity bounds. We illustrate with subgradient algorithms for $p$-mean problems in general Hadamard spaces, in particular computing medians in BHV tree space.

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arXiv2023

Basic convex analysis in metric spaces with bounded curvature

Differentiable structure ensures that many of the basics of classical convex analysis extend naturally from Euclidean space to Riemannian manifolds. Without such structure, however, extensions are more challenging. Nonetheless, in Alexandrov spaces with curvature bounded above (but possibly positive), we develop several basic building blocks. We define subgradients via projection and the normal cone, prove their existence, and relate them to the classical affine minorant property. Then, in what amounts to a simple calculus or duality result, we develop a necessary optimality condition for minimizing the sum of two convex functions.

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arXiv2022

Local linear convergence of alternating projections in metric spaces with bounded curvature

We consider the popular and classical method of alternating projections for finding a point in the intersection of two closed sets. By situating the algorithm in a metric space, equipped only with well-behaved geodesics and angles (in the sense of Alexandrov), we are able to highlight the two key geometric ingredients in a standard intuitive analysis of local linear convergence. The first is a transversality-like condition on the intersection; the second is a convexity-like condition on one set: "uniform approximation by geodesics."

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arXiv2020

A uniform betweenness property in metric spaces and its role in the quantitative analysis of the "Lion-Man" game

In this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a uniform betweenness property and use it in the study of a discrete lion and man game with an $\varepsilon$-capture criterion. In particular, we prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.

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arXiv2019

Geodesic rays, the "Lion-Man" game, and the fixed point property

This paper focuses on the relation among the existence of different types of curves (such as directional ones, quasi-geodesic or geodesic rays), the (approximate) fixed point property for nonexpansive mappings, and a discrete lion and man game. Our main result holds in the setting of CAT(0) spaces that are additionally Gromov hyperbolic.

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arXiv2019

"Lion-Man" and the Fixed Point Property

This paper focuses on the relation between the fixed point property for continuous mappings and a discrete lion and man game played in a strongly convex domain. Our main result states that in locally compact geodesic spaces, the compactness of the domain is equivalent to its fixed point property, as well as to the success of the lion. The common link among these properties involves the existence of different types of rays, which we also discuss.

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arXiv2018

Moduli of regularity and rates of convergence for Fejér monotone sequences

In this paper we introduce the concept of modulus of regularity as a tool to analyze the speed of convergence, including the finite termination, for classes of Fejér monotone sequences which appear in fixed point theory, monotone operator theory, and convex optimization. This concept allows for a unified approach to several notions such as weak sharp minima, error bounds, metric subregularity, Hölder regularity, etc., as well as to obtain rates of convergence for Picard iterates, the Mann algorithm, the proximal point algorithm and the cyclic algorithm. As a byproduct we obtain a quantitative version of the well-known fact that for a convex lower semi-continuous function the set of minimizers coincides with the set of zeros of its subdifferential and the set of fixed points of its resolvent.

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arXiv2014

The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings

In this paper we provide a unified treatment of some convex minimization problems, which allows for a better understanding and, in some cases, improvement of results in this direction proved recently in spaces of curvature bounded above. For this purpose, we analyze the asymptotic behavior of compositions of finitely many firmly nonexpansive mappings in the setting of $p$-uniformly convex geodesic spaces focusing on asymptotic regularity and convergence results.

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