Hadamard Manifolds Research Articles

Penalty function method for a variational inequality on Hadamard manifolds

Penalty function method for a variational inequality on Hadamard manifolds

On the geometry in the large of Hadamard manifolds

A Hadamard manifold is a simply connected, complete Riemannian manifold with nonpositive sectional curvature. The theory of Hadamard manifolds is a topic that has been more and more intensively studied for more than forty years. In the present paper, we prove Liouville-type theorems for conformal, isometric and harmonic pointwise transformations of metrics of Hadamard manifolds.

Existence Results for Generalized Vector Quasi-Equilibrium Problems in Hadamard Manifolds

The purpose of this article was to establish manifold versions of existence theorems for generalized vector quasi-equilibrium problems in locally compact and σ-compact spaces without any continuity assumption. The fixed-point theorem in a product Hadamard manifold is the key focus of our discussion. We further applied our theorems to saddle point and minimax problems.

A projection algorithm for pseudomonotone vector fields with convex constraints on Hadamard manifolds

A projection algorithm for pseudomonotone vector fields with convex constraints on Hadamard manifolds

Mixed Variational Inequality Interval-valued Problem: Theorems of Existence of Solutions

In this article, our efforts focus on finding the conditions for the existence of solutions of Mixed Stampacchia Variational Inequality Interval-valued Problem on Hadamard manifolds with monotonicity assumption by using KKM mappings. Conditions that allow us to prove the existence of equilibrium points in a market of perfect competition. We will identify solutions of Stampacchia variational problem and optimization problem with the interval-valued convex objective function, improving on previous results in the literature. We will illustrate the main results obtained with some examples and numerical results.

On the existence and approximation of solutions of generalized equilibrium problem on Hadamard manifolds

In this paper, we study the existence of solution of the generalized equilibrium problem (GEP) in the framework of an Hadamard manifold. Using the KKM lemma, we prove the existence of solution of the GEP and give the properties of the resolvent function associated with the problem under consideration. Furthermore, we introduce an iterative algorithm for approximating a common solution of the GEP and a fixed point problem. Using the proposed method, we obtain and prove a strong convergence theorem for approximating a solution of the GEP, which is also a fixed point of a nonexpansive mapping under some mild conditions. We give an application of our convergence result to a solution of the convex minimization problem. To illustrate the convergence of the method, we report some numerical experiments. The result in this paper extends the study of the GEP from the linear settings to the Hadamard manifolds.

Approximating Common Fixed Points of Nonexpansive Mappings on Hadamard Manifolds with Applications

The point of this research is to present a new iterative procedure for approximating common fixed points of nonexpansive mappings in Hadamard manifolds. The convergence theorem of the proposed method is discussed under certain conditions. For the sake of clarity, we provide some numerical examples to support our results. Furthermore, we apply the suggested approach to solve inclusion problems and convex feasibility problems.

A generalized geometric spectral conjugate gradient algorithm for finding zero of a monotone tangent vector field on a constant curvature Hadamard manifold

This paper is concerned with the problem of finding a zero of a monotone tangent vector field on a constant curvature Hadamard manifold. We propose a geometric spectral conjugate gradient method for solving this problem. The global convergence of the proposed method is established under some conditions. Illustrative numerical examples are also presented to show the practical effectiveness of our method.

An inertial proximal point method for difference of maximal monotone vector fields in Hadamard manifolds

An inertial proximal point method for difference of maximal monotone vector fields in Hadamard manifolds

Levitin–Polyak Well-Posedness by Perturbations for the Split Hemivariational Inequality Problem on Hadamard Manifolds

Levitin–Polyak Well-Posedness by Perturbations for the Split Hemivariational Inequality Problem on Hadamard Manifolds

Combinatorial Convexity in Hadamard Manifolds: Existence for Equilibrium Problems

Combinatorial Convexity in Hadamard Manifolds: Existence for Equilibrium Problems

Optimality Conditions for Multiobjective Mathematical Programming Problems with Equilibrium Constraints on Hadamard Manifolds

In this paper, we consider a class of multiobjective mathematical programming problems with equilibrium constraints on Hadamard manifolds (in short, (MMPEC)). We introduce the generalized Guignard constraint qualification for (MMPEC) and employ it to derive Karush–Kuhn–Tucker (KKT)-type necessary optimality criteria. Further, we derive sufficient optimality criteria for (MMPEC) using geodesic convexity assumptions. The significance of the results deduced in the paper has been demonstrated by suitable non-trivial examples. The results deduced in this article generalize several well-known results in the literature to a more general space, that is, Hadamard manifolds, and extend them to a more general class of optimization problems. To the best of our knowledge, this is the first time that generalized Guignard constraint qualification and optimality conditions have been studied for (MMPEC) in manifold settings.

Regularization methods for hierarchical variational inequality problems on Hadamard manifolds

This article proposes two regularized iterative algorithms for solving variational inequality problems defined over a solution set of a variational inclusion problem, known as hierarchical variational inequality problems, in the setting of Hadamard manifolds. Instead of regularizing the variational inequality problem or an iterative method for solving it, we first regularize the considered variational inclusion problem, and then prove that the solution of the regularized problem converges to a solution of the hierarchical variational inequality problem. Using such a result, we prove the convergence of the sequences generated by the proposed algorithms to a solution of the considered hierarchical variational inequality problem. A computational experiment is provided to see the validity and effectiveness of the proposed algorithms.

A Contribution of Liouville-Type Theorems to the Geometry in the Large of Hadamard Manifolds

A complete, simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. In this article, we prove Liouville-type theorems for isometric and harmonic self-diffeomorphisms of Hadamard manifolds, as well as Liouville-type theorems for Killing–Yano, symmetric Killing and harmonic tensors on Hadamard manifolds.

Iterative Approximate Solutions for Variational Problems in Hadamard Manifold

The goal of this paper is to propose and investigate new iterative methods for examining an approximate solution of a fixed-point problem, an equilibrium problem, and a finite collection of variational inclusions in the Hadamard manifold’s structure. Operating under some assumptions, we extend the proximal point algorithm to estimate the common solution of stated problems and obtain a strong convergence theorem for the common solution. We also present several consequences of the proposed iterative methods and their convergence results.

Iterative algorithms for monotone variational inequality and fixed point problems on Hadamard manifolds

Iterative algorithms for monotone variational inequality and fixed point problems on Hadamard manifolds

Optimality conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds using generalized geodesic convexity

This paper deals with multiobjective semi-infinite programming problems on Hadamard manifolds. We establish the sufficient optimality criteria of the considered problem under generalized geodesic convexity assumptions. Moreover, we formulate the Mond-Weir and Wolfe type dual problems and derive the weak, strong and strict converse duality theorems relating the primal and dual problems under generalized geodesic convexity assumptions. Suitable examples have also been given to illustrate the significance of these results. The results presented in this paper extend and generalize the corresponding results in the literature.

Inexact proximal point algorithm for quasiconvex optimization problems on Hadamard manifolds

ABSTRACT In this paper, by using the inexact scalarization proximal methods, we solve quasiconvex multiobjective optimization problems on Hadamard manifolds. Under some assumptions on the multifunction of problem and vector fields, our methods are proved to be convergent to a Pareto critical point of the problem. For the convex case, the sequences generated by the methods converge to a weak Pareto solution.

On Minty variational principle for nonsmooth multiobjective optimization problems on Hadamard manifolds

In this paper, we consider classes of approximate Minty and Stampacchia type vector variational inequalities using Clarke subdifferential on Hadamard manifolds and a class of nonsmooth multiobjective optimization problems. We investigate the relationship between the solution of these approximate vector variational inequalities and the solution of nonsmooth multiobjective optimization problems involving geodesic approximately convex functions. The results presented in this paper extend and generalize some existing results in the literature.

" Implicit and explicit viscosity methods for hierarchical variational inequalities on Hadamard manifolds"

" Implicit and explicit viscosity methods for hierarchical variational inequalities on Hadamard manifolds"