Hadamard Spaces Research Articles

Two-Step Proximal Method for Equilibrium Problems in Hadamard spaces

We propose a novel two-step proximal method for solving equilibrium problems in Hadamard spaces. The equilibrium problem is very general in the sense that it includes as special cases many applied mathematical models such as: variational inequalities, optimization problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithm is the analog of the two-step algorithm for solving the equilibrium problem in Hilbert spaces explored earlier. We prove the weak convergence of the sequence generated by the algorithm for pseudo-monotone bifunctions. Our results extend some known results in the literature for pseudo-monotone equilibrium problems.

The viscosity approximation method for multivalued G-nonexpansive mappings in Hadamard spaces endowed with graphs

Using the viscosity approximation method introduced by Moudafi (J Math Anal Appl 241:46–55, 2000), we can obtain strong convergence theorems for monotone increasing G-nonexpansive mappings in Hadamard spaces endowed with graphs. We also give sufficient conditions for the existence of solutions of the variational inequality problem in this setting. Our results generalize and improve many results in the literature.

Convergence of a Two-Stage Proximal Algorithm for the Equilibrium Problem in Hadamard Spaces

An iterative two-stage proximal algorithm for approximate solution of equilibrium problems in Hadamard spaces is considered. This algorithm is an analog of the already studied two-stage algorithm for equilibrium problems in a Hilbert space. For Lipschitz-type pseudo-monotone bifunctions, a theorem on the weak convergence of sequences generated by the algorithm is proved.

Adaptive algorithms for equilibrium problems in Hadamard spaces

Adaptive algorithms for equilibrium problems in Hadamard spaces

On θ-generalized demimetric mappings and monotone operators in Hadamard spaces

AbstractOur main interest in this article is to introduce and study the class of θ-generalized demimetric mappings in Hadamard spaces. Also, a Halpern-type proximal point algorithm comprising this class of mappings and resolvents of monotone operators is proposed, and we prove that it converges strongly to a fixed point of a θ-generalized demimetric mapping and a common zero of a finite family of monotone operators in a Hadamard space. Furthermore, we apply the obtained results to solve a finite family of convex minimization problems, variational inequality problems and convex feasibility problems in Hadamard spaces.

Fitzpatrick transform of monotone relations in Hadamard spaces

Abstract In the present paper, monotone relations and maximal monotone relations from an Hadamard space to its linear dual space are investigated. Fitzpatrick transform of monotone relations in Hadamard spaces is introduced. It is shown that Fitzpatrick transform of a special class of monotone relations is proper, convex and lower semi-continuous. Finally, a representation result for monotone relations is given.

A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces

The purpose of this paper is to propose and analyze a multi-step iterative sequence to solve a convex optimization problem and a fixed point problem in an Hadamard space. We aim to establish strong and $$ \triangle $$ -convergence results of the proposed iterative sequence by employing suitable conditions on the control parameters and the structural properties of the underlying space. As a consequence, we compute an optimal solution for a minimizer of proper convex lower semicontinuous function and a common fixed point of a finite family of total asymptotically quasi-nonexpansive mappings in Hadamard spaces. Our results can be viewed as an extension and generalization of various corresponding results in the existing literature.

An extragradient algorithm for solving equilibrium problem and zero point problem in Hadamard spaces

In this article, using a hybrid extragradient method, we introduce a new iterative process for approximating a common element of the set of solutions of an equilibrium problem and a common zero of a finite family of monotone operators in Hadamard spaces. We also give a numerical example to solve a nonconvex optimization problem in an Hadamard space to support our main result.

Convergence theorems for asymptotically quasi-nonexpansive sequences with applications

In this paper, -convergence and strong convergence theorems for asymptotically quasi-nonexpansive sequences in Hadamard spaces are derived. The results extend and improve recent results in the literature and some of them seem new even in Hilbert spaces. Applications to convex minimization and common fixed point problems are discussed.

A viscosity-type proximal point algorithm for monotone equilibrium problem and fixed point problem in an Hadamard space

In this paper, we introduce a viscosity-type proximal point algorithm comprising of a finite composition of resolvents of monotone bifunctions and a generalized asymptotically nonspreading mapping recently introduced by Phuengrattana [Appl. Gen. Topol. 18 (2017) 117–129]. We establish a strong convergence result of the proposed algorithm to a common solution of a finite family of equilibrium problems and fixed point problem for a generalized asymptotically nonspreading and nonexpansive mappings, which is also a unique solution of some variational inequality problems in an Hadamard space. We apply our result to solve convex feasibility problem and to approximate a common solution of a finite family of minimization problems in an Hadamard space.

On homotopy types of Vietoris–Rips complexes of metric gluings

Given a sample of points $X$ in a metric space $M$ and a scale $r>0$, the Vietoris-Rips simplicial complex $\mathrm{VR}(X;r)$ is a standard construction to attempt to recover $M$ from $X$ up to homotopy type. A deficiency of this approach is that $\mathrm{VR}(X;r)$ is not metrizable if it is not locally finite, and thus does not recover metric information about $M$. We attempt to remedy this shortcoming by defining a metric space thickening of $X$, which we call the \emph{Vietoris-Rips thickening} $\mathrm{VR}^m(X;r)$, via the theory of optimal transport. When $M$ is a complete Riemannian manifold, or alternatively a compact Hadamard space, we show that the the Vietoris-Rips thickening satisfies Hausmann's theorem ($\mathrm{VR}^m(M;r)\simeq M$ for $r$ sufficiently small) with a simpler proof: homotopy equivalence $\mathrm{VR}^m(M;r)\to M$ is canonically defined as a center of mass map, and its homotopy inverse is the (now continuous) inclusion map $M\hookrightarrow\mathrm{VR}^m(M;r)$. Furthermore, we describe the homotopy type of the Vietoris-Rips thickening of the $n$-sphere at the first positive scale parameter $r$ where the homotopy type changes.

Fixed Points of Multivalued G-Nonexpansive Mappings in Hadamard Spaces Endowed with Graphs

A fixed-point theorem for multivalued G-nonexpansive mappings in Hadamard spaces is proved. The strong convergence theorems of Browder’s and Halpern’s iterations are also established. Our results generalize and improve the results of Tiammee et al. (2015), Alfuraidan and Khamsi (2015), Anakkamatee and Tongnoi (2019), and many others.

Almost periodicity and ergodic theorems for nonexpansive mappings and semigroups in Hadamard spaces

The main purpose of this paper is to prove the mean ergodic theorem for nonexpansive mappings and semigroups in locally compact Hadamard spaces, including finite dimensional Hadamard manifolds. The main tool for proving ergodic convergence is the almost periodicity of orbits of a nonexpansive mapping. Therefore, in the first part of the paper, we study almost periodicity (and as a special case, periodicity) in metric and Hadamard spaces. Then, we prove a mean ergodic theorem for nonexpansive mappings and continuous semigroups of contractions in locally compact Hadamard spaces. Finally, an application to the asymptotic behavior of the first order evolution equation associated to the monotone vector field on Hadamard manifolds is presented.

A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space

In this paper, we introduce a viscosity-type proximal point algorithm which comprises of a finite sum of resolvents of monotone operators, and a generalized asymptotically nonexpansive mapping. We prove that the algorithm converges strongly to a common zero of a finite family of monotone operators, which is also a fixed point of a generalized asymptotically nonexpansive mapping in an Hadamard space. Furthermore, we give two numerical examples of our algorithm in finite dimensional spaces of real numbers and one numerical example in a non-Hilbert space setting, in order to show the applicability of our results.

Weak sharpness and finite termination for variational inequalities on Hadamard manifolds

We first introduce the notion of weak sharpness for the solution sets of variational inequality problems (in short, VIP) on Hadamard spaces. We then study the finite convergence property of sequences generated by the inexact proximal point algorithm with different error terms for solving VIP under weak sharpness of the solution set. We also give an upper bound on the number of iterations by which the sequence generated by the exact proximal point algorithm converges to a solution of VIP. An example is also given to illustrate our results.

Tauberian Conditions for Almost Convergence in a Geodesic Metric Space

In the present paper, after recalling the Karcher mean in Hadamard spaces, we study the relation between convergence, almost convergence and mean convergence (respect to the defined mean) of a sequence in Hadamard spaces. These results extend Tauberian conditions from Banach spaces to Hadamard spaces. Also, we show that every almost periodic sequence in Hadamard spaces is almost convergent.

Convergence analysis of the extragradient method for equilibrium problems in Hadamard spaces

We propose an extragradient method for solving equilibrium problems of pseudo-monotone type in Hadamard spaces. We prove $$\Delta $$-convergence of the generated sequence to a solution of the equilibrium problem, under standard assumptions on the bifunction. Then, we propose a regularization procedure which ensures strong convergence of the generated sequence to an equilibrium point of the problem.

Viscosity approximation methods for generalized asymptotically nonexpansive mappings and monotone inclusions in Hadamard spaces

Viscosity approximation methods for generalized asymptotically nonexpansive mappings and monotone inclusions in Hadamard spaces

A two-stage proximal algorithm for equilibrium problems in Hadamard spaces

A two-stage proximal algorithm for equilibrium problems in Hadamard spaces

Common solution for a finite family of minimization problem and fixed point problem for a pair of demicontractive mappings in Hadamard spaces

In this paper, an iterative algorithm to approximate a common solution of a finite family of minimization problem and fixed point problem for a pair of demicontractive mappings in Hadamard spaces are proposed. Under suitable conditions, some $$\Delta $$-convergence and strong convergence theorems of the sequence generated by the algorithm to an element in the intersection of the set of solutions of such kind of minimization problem and fixed point problem in Hardmard space are proved. Our results complement and extend some recent results in literature.