Nonexpansive mappings and monotone vector fields in Hadamard manifolds

This paper briefly surveys some recent advances in the investigation of nonexpansive mappings and monotone vector fields, focusing in the extension of basic results of the classical nonlinear functional analysis from Banach spaces to the class of nonpositive sectional curvature Riemannian manifolds called Hadamard manifolds. Within this setting, we first analyze the problem of finding fixed points of nonexpansive mappings. Later on, different classes of monotonicity for set-valued vector fields and the relationship between some of them will be presented, followed by the study of the existence and approximation of singularities for such vector fields. We will discuss about variational inequality and minimization problems in Hadamard manifolds, stressing the fact that these problems can be solved by means of the iterative approaches for monotone vector fields

Bregman strongly nonexpansive operators in reflexive Banach spaces

We present a detailed study of right and left Bregman strongly nonexpansive operators in reflexive Banach spaces. We analyze, in particular, compositions and convex combinations of such operators, and prove the convergence of the Picard iterative method for operators of these types. Finally, we use our results to approximate common zeroes of maximal monotone mappings and solutions to convex feasibility problems.Ministerio de Educación y CienciaJunta de AndalucíaIsrael Science Foundatio

Right Bregman nonexpansive operators in Banach spaces

We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.Dirección General de Enseñanza SuperiorJunta de AndalucíaIsrael Science FoundationGraduate School of the TechnionFund for the Promotion of Research at the TechnionTechnion VPR Fun

Iterative methods for approximating fixed points of Bregman nonexpansive operators

Diverse notions of nonexpansive type operators have been extended to the more general framework of Bregman distances in reflexive Banach spaces. We study these classes of operators, mainly with respect to the existence and approximation of their (asymptotic) fixed points. In particular, the asymptotic behavior of Picard and Mann type iterations is discussed for quasi-Bregman nonexpansive operators. We also present parallel algorithms for approximating common fixed points of a finite family of Bregman strongly nonexpansive operators by means of a block operator which preserves the Bregman strong nonexpansivity. All the results hold, in particular, for the smaller class of Bregman firmly nonexpansive operators, a class which contains the generalized resolvents of monotone mappings with respect to the Bregman distance.Dirección General de Enseñanza SuperiorJunta de AndalucíaIsrael Science FoundationGraduate School of the TechnionFund for the Promotion of Research at the TechnionTechnion President’s Research Fun

Iterative algorithms for nonexpansive mappings on Hadamard manifolds

Two iterative algorithms for nonexpansive mappings on Hadamard manifolds, which are extensions of the well-known Halpern's and Mann's algorithms in Euclidean spaces, are proposed and proved to be convergent to a fixed point of the mapping. Some numerical examples are provided.Dirección General de Enseñanza SuperiorNational Natural Science Foundations of ChinaJunta de AndalucíaMinisterio de Ciencia e Innovació

Monotone and accretive vector fields on Riemannian manifolds

The relationship between monotonicity and accretivity on Riemannian manifolds is studied in this paper and both concepts are proved to be equivalent in Hadamard manifolds. As a consequence an iterative method is obtained for approximating singularities of Lipschitz continuous, strongly monotone mappings. We also establish the equivalence between the strong convexity of convex functions and the strong monotonicity of its subdifferentials on Riemannian manifolds. These results are then applied to solve the minimization problem of convex functions on Riemannian manifolds.Ministerio de Ciencia e InnovaciónJunta de AndalucíaNational Natural Science Foundations of Chin

Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular

Because of Minty’s classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or “almost have” fixed points, then the same is true for their composition. This significantly generalizes the result by Bauschke from 2003 for the case of projectors (nearest point mappings). The proof resides in a Hilbert product space and it relies upon the Brezis-Haraux range approximation result. By working in a suitably scaled Hilbert product space, we also establish the asymptotic regularity of convex.Natural Sciences and Engineering Research Council of CanadaCanada Research Chair ProgramDirección General de Enseñanza SuperiorJunta de Andaluci

Forward-Backward splitting methods for accretive operators in Banach spaces

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce two iterative forward-backward splitting methods with relaxations and errors to find zeros of the sum of two accretive operators in the Banach spaces. We shall prove the weak and strong convergence of these methods under mild conditions. We also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem

Applications of convex analysis within mathematics

This talk is based on the paper: Aragón, Borwein, Martín-Márquez, Yao Applications of convex analysis within mathematics, Math. Program., Ser B, December 2014, Volume 148, Issue 1, pp 49-88. in a special issue to celebrate the 50th birthday of Modern Convex Analysis and convex optimization that became a tribute to the memory of Jean Jacques Moreau who passed away (on January 9, 2014) as the edition was being completed

Phraseology and oral discourse in the TV Series La que se avecina

Las series televisivas ejercen una influencia significativa en sus consumidores, especialmente en lo que atañe a estereotipos lingüísticos y, por lo tanto, identitarios. En este contexto, este trabajo estudia la fraseología en la ficción seriada La que se avecina, actualmente en emisión y que cuenta con un considerable número de seguidores. Para ello, además de un marco teórico, se ha partido del concepto de pragmatema, que hace referencia a las expresiones lingüísticas fraseológicas que están restringidas por situaciones extralingüísticas particulares en su uso. El análisis de este eficaz y común elemento de comunicación a través del entorno seriado televisivo supone reconocer el modelo de comunicación empleado por este medio en la actualidad y el comportamiento de sus consumidores. Metodológicamente, tratamos de averiguar qué tipo de frases hechas o pragmatemas particulares de distintos personajes de la misma se han establecido en la lengua coloquial de los receptores a partir de una encuesta online de Google publicada en las redes sociales Twitter, Instagram, Facebook y WhatsApp, de manera que los resultados obtenidos pudieran ser anónimos, espontáneos e inmediatos. Los resultados concluyen en que estos elementos lingüísticos se han establecido en la lengua coloquial de nuestra sociedad.Television series exert considerable influence on their consumers, especially as regards linguistic and identity stereotypes. This paper studies the phraseology in the fiction series La que se avecina, which is currently on air and has a large number of viewers. Apart from a theoretical framework, the concept of pragmatemes has been used, which refers to phraseological linguistic expressions, use of which is restricted by particular extralinguistic situations. The analysis of this effective and common element of communication through television series involves identifying the communication model currently utilised by the medium and its consumers’ behaviour. Methodologically, we have strived to look at the type of set phrases or particular pragmatemes employed by different characters in the series which have entered the receivers’ colloquial speech. A Google online survey was posted on the social media platforms Twitter, Instagram, Facebook and WhatsApp to ensure that the results are anonymous, spontaneous and immediate. The results show that these linguistic elements have been established in Spanish society’s colloquial languag