New Characterization of Geodesic Convexity on Hadamard Manifolds with Applications

In this paper, some new results, concerned with the geodesic convex hull and geodesic convex combination, are given on Hadamard manifolds. An S-KKM theorem on a Hadamard manifold is also given in order to generalize the KKM theorem. As applications, a Fan–Browder-type fixed point theorem and a fixed point theorem for the a new mapping class are proved on Hadamard manifolds.     

Rate of convergence for proximal point algorithms on Hadamard manifolds

In this paper, an estimate of convergence rate concerned with an inexact proximal point algorithm for the singularity of maximal monotone vector fields on Hadamard manifolds is discussed. We introduce a weaker growth condition, which is an extension of that of Luque from Euclidean spaces to Hadamard manifolds. Under the growth condition, we prove that the inexact proximal point algorithm has linear/superlinear convergence rate. The main results presented in this paper generalize and improve some corresponding known results.     

Generalized Browder-type fixed point theorem with strongly geodesic convexity on Hadamard manifolds with applications

In this paper, a generalized Browder-type fixed point theorem on Hadamard manifolds is introduced, which can be regarded as a generalization of the Browder-type fixed point theorem for the set-valued mapping on an Euclidean space to a Hadamard manifold. As applications, a maximal element theorem, a section theorem, a Ky Fan-type Minimax Inequality and an existence theorem of Nash equilibrium for non-cooperative games on Hadamard manifolds are established.     

Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications

In this paper, maximal element theorem on Hadamard manifolds is established. First, we prove the existence of solutions for maximal element theorem on Hadamard manifolds. Further, we prove that most of problems in maximal element theorem on Hadamard manifolds (in the sense of Baire category) are essential and that, for any problem in maximal element theorem on Hadamard manifolds, there exists at least one essential component of its solution set. As applications, we study existence and stability of solutions for variational relation problems on Hadamard manifolds, and existence and stability of weakly Pareto-Nash equilibrium points for n-person multi-objective games on Hadamard manifolds.     

Iteration-Complexity of Gradient,Subgradient and Proximal Point Methods on Riemannian Manifolds

This paper considers optimization problems on Riemannian manifolds and analyzes the iteration-complexity for gradient and subgradient methods on manifolds with nonnegative curvatures. By using tools from Riemannian convex analysis and directly exploring the tangent space of the manifold, we obtain different iteration-complexity bounds for the aforementioned methods, thereby complementing and improving related results. Moreover, we also establish an iteration-complexity bound for the proximal point method on Hadamard manifolds.     

An inexact proximal point algorithm for maximal monotone vector fields on Hadamard manifolds

In this paper, an inexact proximal point algorithm concerned with the singularity of maximal monotone vector fields is introduced and studied on Hadamard manifolds, in which a relative error tolerance with squared summable error factors is considered. It is proved that the sequence generated by the proposed method is convergent to a solution of the problem. Moreover, an application to the optimization problem on Hadamard manifolds is given. The main results presented in this paper generalize and improve some corresponding known results given in the literature.     

A Cartan–Hadamard Theorem for Banach–Finsler Manifolds

In this paper we study Banach–Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive differential in every point. In this context we generalize the classical theorem of Cartan–Hadamard, saying that the exponential function is a covering map. We apply this to symmetric spaces and thus obtain criteria for Banach–Lie groups with an involution to have a polar decomposition. Typical examples of symmetric Finsler manifolds with seminegative curvature are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure which in general is not Riemannian.     

Halpern Iteration in CAT（k） Spaces

In this paper we show that an iterative sequence generated by the Halpern algorithm converges to a fixed point in the case of complete CAT(κ) spaces. Similar results for Hadamard manifolds were obtained in [Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math., 14, 541–559 (2010)], but we study a much more general case. Moreover, we discuss the Halpern iteration procedure for set-valued mappings.     

Halpern- and Mann-Type Algorithms for Fixed Points and Inclusion Problems on Hadamard Manifolds

In this article, we consider an inclusion problem which is defined by means of a sum of a single-valued vector field and a set-valued vector field defined on a Hadamard manifold. We propose Halpern-type and Mann-type algorithms for finding a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. Some particular cases of our problem and algorithm are also discussed. We study the convergence of the proposed algorithm to a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. As applications of our results and algorithms, we derive the solution methods and their convergence results for the optimization problems, variational inequality problems and equilibrium problems in the setting of Hadamard manifolds.     

An extension of the proximal point algorithm with Bregman distances on Hadamard manifolds

In this paper we present an extension of the proximal point algorithm with Bregman distances to solve constrained minimization problems with quasiconvex and convex objective function on Hadamard manifolds. The proposed algorithm is a modified and extended version of the one presented in Papa Quiroz and Oliveira (J Convex Anal 16(1): 49–69, 2009). An advantage of the proposed algorithm, for the nonconvex case, is that in each iteration the algorithm only needs to find a stationary point of the proximal function and not a global minimum. For that reason, from the computational point of view, the proposed algorithm is more practical than the earlier proximal method. Another advantage, for the convex case, is that using minimal condition on the problem data as well as on the proximal parameters we get the same convergence results of the Euclidean proximal algorithm using Bregman distances.     

New results on the proximal point algorithm in non-positive curvature metric spaces

The subject of this paper is the inexact proximal point algorithm of usual and Halpern type in non-positive curvature metric spaces. We study the convergence of the sequences given by the inexact proximal point algorithm with non-summable errors. We also prove the strong convergence of the Halpern proximal point algorithm to a minimum point of the convex function. The results extend several results in Hilbert spaces, Hadamard manifolds and non-positive curvature metric spaces.     

Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds

This paper is focused on the problem of finding a singularity of the sum of two vector fields defined on a Hadamard manifold, or more precisely, the study of a generalized equation in a Riemannian setting. We extend the concept of metric regularity to the Riemannian setting and investigate its relationship with the generalized equation in this new context. In particular, a version of Graves’s theorem is presented and we also define some concepts related to metric regularity, including the Aubin property and the strong metric regularity of set-valued vector fields. A conceptual method for finding a singularity of the sum of two vector fields is also considered. This method has as particular instances: the proximal point method, Newton’s method, and Zincenko’s method on Hadamard manifolds. Under the assumption of metric regularity at the singularity, we establish that the methods are well defined in a suitable neighborhood of the singularity. Moreover, we also show that each sequence generated by these methods converges to this singularity at a superlinear rate.     

Existence of non-trivial harmonic functions on Cartan–Hadamard manifolds of unbounded curvature

The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) attracted a lot of attention. For Cartan–Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan–Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan–Hadamard manifolds is much more complicated than one might have expected.      

Dynamical systems for solving variational inclusion and fixed point problems on Hadamard manifolds

In this paper, we investigate a new class of dynamical systems for solving variational inclusion and fixed point problems on Hadamard manifolds. Then we prove that the dynamical system has a unique solution under some suitable assumptions. Moreover, the global exponential stability and invariance property of the dynamical systems are also established. Our main results in this work are new and extend the existing ones in the literature.     

The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds

In this paper, we investigate the proximal point algorithm (in short PPA) for variational inequalities with pseudomonotone vector fields on Hadamard manifolds. Under weaker assumptions than monotonicity, we show that the sequence generated by PPA is well defined and prove that the sequence converges to a solution of variational inequality, whenever it exists. The results presented in this paper generalize and improve some corresponding known results given in literatures.     

Korpelevich’s method for variational inequality problems on Hadamard manifolds

The concept of pseudomonotone vector field on Hadamard manifold is introduced. A variant of Korpelevich??s method for solving the variational inequality problem is extended from Euclidean spaces to constant curvature Hadamard manifolds. Under a pseudomonotone assumption on the underlying vector field, we prove that the sequence generated by the method converges to a solution of variational inequality, whenever it exists. Moreover, we give an example to show the effectiveness of our method.     

Proximal Point Algorithm for Quasi-Convex Minimization Problems in Metric Spaces

In this paper, the proximal point algorithm for quasi-convex minimization problem in nonpositive curvature metric spaces is studied. We prove Δ-convergence of the generated sequence to a critical point (which is defined in the text) of an objective quasi-convex, proper and lower semicontinuous function with at least a minimum point as well as some strong convergence results to a minimum point with some additional conditions. The results extend the recent results of the proximal point algorithm in Hadamard manifolds and CAT(0) spaces.     

Vector variational inequality with pseudoconvexity on Hadamard manifolds

In this paper, we give some properties for nondifferentiable pseudoconvex functions on Hadamard manifolds, and discuss the connections between pseudoconvex functions and pseudomonotone vector fields. Moreover, we study Minty and Stampacchia vector variational inequalities, which are formulated in terms of Clarke subdifferential for nonsmooth functions. Some relations between the vector variational inequalities and nonsmooth vector optimization problems are established under pseudoconvexity or pseudomonotonicity. The results presented in this paper extend some corresponding known results given in the literatures.     

Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds

In this paper, we study eigenvalues of elliptic operators in divergence form on compact Riemannian manifolds with boundary (possibly empty) and obtain a general inequality for them. By using this inequality, we prove universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds in a Euclidean space, and of complete manifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifold and manifolds admitting eigenmaps to a sphere.     

On the Asymptotic Geometry of Nonpositively Curved Manifolds

In this paper we derive new asymptotic properties of all Hadamard manifolds admitting compact quotients. We study the growth function of the volume of geodesic spheres, generalizing the work of Margulis in the case of negative curvature. We show that the growth is of purely exponential type if and only if the Hadamard manifold is of rank 1. In general, there is a polynomial deviation from purely exponential behavior, depending in an unexpected way on the rank of the manifold. Furthermore, we obtain new results on the growth rate of closed geodesics on compact rank 1 spaces. Submitted: August 1996, final version: April 1997