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

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

Riemannian stochastic approximation algorithms

We examine a wide class of stochastic approximation algorithms for solving (stochastic) nonlinear problems on Riemannian manifolds. Such algorithms arise naturally in the study of Riemannian optimization, game theory and optimal transport, but their behavior is much less understood compared to the Euclidean case because of the lack of a global linear structure on the manifold. We overcome this difficulty by introducing a suitable Fermi coordinate frame which allows us to map the asymptotic behavior of the Riemannian Robbins-Monro (RRM) algorithms under study to that of an associated deterministic dynamical system. In so doing, we provide a general template of almost sure convergence results that mirrors and extends the existing theory for Euclidean Robbins-Monro schemes, despite the significant complications that arise due to the curvature and topology of the underlying manifold. We showcase the flexibility of the proposed framework by applying it to a range of retraction-based variants of the popular optimistic / extra-gradient methods for solving minimization problems and games, and we provide a unified treatment for their convergence.Comment: 33 pages, 2 figures; a one-page abstract of this paper was presented in COLT 202

Equilibrium problems on Riemannian manifolds with applications

We study the equilibrium problem on general Riemannian manifolds. The results on existence of solutions and on the convex structure of the solution set are established. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. As applications, the corresponding results for the mixed variational inequality and the Nash equilibrium are obtained. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388, 61-77, 2012 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on Hadamard manifolds.National Natural Science Foundation of ChinaNatural Science Foundation of Guizhou Province (China)Dirección General de Enseñanza SuperiorJunta de AndalucíaNational Science Council of Taiwa

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

Nonlinear Analysis and Optimization with Applications

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world