Hadamard Manifolds Research Articles

Pareto Efficiency Criteria and Duality for Multiobjective Fractional Programming Problems with Equilibrium Constraints on Hadamard Manifolds

This article deals with multiobjective fractional programming problems with equilibrium constraints in the setting of Hadamard manifolds (abbreviated as MFPPEC). The generalized Guignard constraint qualification (abbreviated as GGCQ) for MFPPEC is presented. Furthermore, the Karush–Kuhn–Tucker (abbreviated as KKT) type necessary criteria of Pareto efficiency for MFPPEC are derived using GGCQ. Sufficient criteria of Pareto efficiency for MFPPEC are deduced under some geodesic convexity hypotheses. Subsequently, Mond–Weir and Wolfe type dual models related to MFPPEC are formulated. The weak, strong, and strict converse duality results are derived relating MFPPEC and the respective dual models. Suitable nontrivial examples have been furnished to demonstrate the significance of the results established in this article. The results derived in the article extend and generalize several notable results previously existing in the literature. To the best of our knowledge, optimality conditions and duality for MFPPEC have not yet been studied in the framework of manifolds.

On the Spectrum of Certain Hadamard Manifolds

We show the absolute continuity of the spectrum and determine the spectrum as a set for two classes of Hadamard manifolds and for specific domains and quotients of one of the classes.

Linear isoperimetric inequality for homogeneous Hadamard manifolds

Linear isoperimetric inequality for homogeneous Hadamard manifolds

New subgradient extragradient algorithm for solving variational inequalities in Hadamard manifold

This paper introduces a schematic approximation method for a solution to a variational inequality problem in the framework of Hadamard manifold. The method is a combination of the subgradient extragradient technique and Popov extragradient method. Using this method, some convergence algorithms were proved when the cost operators are pseudomonotone and strongly pseudomonotone, respectively. In the construction of this method, the dependence on Lipschitz constants of the operators is dispensed with by the use of a monotone decreasing step size. We give an application of our main result to the constrained convex minimization problem. Finally, we report some numerical examples to illustrate the efficiency and applicability of the method.

A Novel Halpern-type Algorithm for a Monotone Inclusion Problem and a Fixed Points Problem on Hadamard Manifolds

In this paper, we propose a novel Halpern-type algorithm and prove that the sequence generated by the algorithm converges strongly to the common element of the set of fixed points of the two firmly nonexpansive mappings and the solution set of zero points of the monotone inclusion problems on Hadamard manifolds, the main results in this paper extended and improved some recent related results.

Second-Order Optimality Conditions and Duality for Multiobjective Semi-Infinite Programming Problems on Hadamard Manifolds

This paper is devoted to the study of multiobjective semi-infinite programming problems on Hadamard manifolds. We consider a class of multiobjective semi-infinite programming problems (abbreviated as MSIP) on Hadamard manifolds. We use the concepts of second-order Karush–Kuhn–Tucker stationary point and second-order Karush–Kuhn–Tucker geodesic pseudoconvexity of the considered problem to derive necessary and sufficient second-order conditions of efficiency, weak efficiency and proper efficiency for MSIP along with certain generalized geodesic convexity assumptions. Moreover, we formulate the second-order Mond–Weir-type dual problem related to MSIP and deduce weak and strong duality theorems relating MSIP and the dual problem. The significance of our results is demonstrated with the help of non-trivial examples. To the best of our knowledge, this is the first time that second-order optimality conditions for MSIP have been studied in Hadamard manifold setting.

Minkowski Inequality in Cartan–Hadamard Manifolds

Abstract Using harmonic mean curvature flow, we establish a sharp Minkowski-type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard $3$-manifolds. This inequality also improves the known estimates for total mean curvature in hyperbolic $3$-space. As an application, we obtain a Bonnesen-style isoperimetric inequality for surfaces with convex distance function in nonpositively curved $3$-spaces, via monotonicity results for total mean curvature. This connection between the Minkowski and isoperimetric inequalities is extended to Cartan–Hadamard manifolds of any dimension.

Optimality Conditions and Duality for Nonsmooth Multiobjective Semi-infinite Programming Problems on Hadamard Manifolds

Optimality Conditions and Duality for Nonsmooth Multiobjective Semi-infinite Programming Problems on Hadamard Manifolds

A Riemannian Inertial Mann Algorithm for Nonexpansive Mappings on Hadamard Manifolds

A Riemannian Inertial Mann Algorithm for Nonexpansive Mappings on Hadamard Manifolds

Nonsmooth vector variational inequalities on Hadamard manifold and their existence

Nonsmooth vector variational inequalities on Hadamard manifold and their existence

A new self-adaptive iterative method for variational inclusion problems on Hadamard manifolds with applications

The objective of this work is to design a new iterative method based on Armijo’s type-modified extragradient method for solving the inclusion problem $$\varvec{(A+B)^{-1}(0)}$$ , where $$\varvec{A}$$ is a maximal monotone vector field and $$\varvec{B}$$ is a continuous monotone vector field. The proposed method requires one projection at each iteration, reducing the cost of computational viewpoint and improving the convergence rate. A convergence theorem is established for the proposed extragradient method, significantly improving existing results. We provide concrete examples of Hadamard manifolds and convergency for numerical confirmation. Moreover, we demonstrate convergence results for the variational inequality problems in which the vector field’s monotonicity can be removed.

Duality for Multiobjective Programming Problems with Equilibrium Constraints on Hadamard Manifolds under Generalized Geodesic Convexity

This article is devoted to the study of a class of multiobjective mathematical programming problems with equilibrium constraints on Hadamard manifolds (in short, (MPPEC)). We consider (MPPEC) as our primal problem and formulate two different kinds of dual models, namely, Wolfe and Mond-Weir type dual models related to (MPPEC). Further, we deduce the weak, strong as well as strict converse duality relations that relate (MPPEC) and the corresponding dual problems employing geodesic pseudoconvexity and geodesic quasiconvexity restrictions. Several suitable numerical examples are incorporated to demonstrate the significance of the deduced results. The results derived in this article generalize and extend several previously existing results in the literature.

Nonsmooth interval-valued multiobjective optimization problems and generalized variational inequalities on Hadamard manifolds

Nonsmooth interval-valued multiobjective optimization problems and generalized variational inequalities on Hadamard manifolds

On Constraint Qualifications for Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds

On Constraint Qualifications for Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds

Harmonic quasi-isometric maps III: quotients of Hadamard manifolds

In a previous paper, we proved that a quasi-isometric map $$f:X\rightarrow Y$$ between two pinched Hadamard manifolds X and Y is within bounded distance from a unique harmonic map. We extend this result to maps $$f:\Gamma \backslash X\rightarrow Y$$ , where $$\Gamma $$ is a convex cocompact discrete group of isometries of X and f is locally quasi-isometric at infinity.

Peano property, two dimensional triangles, constant curvature

In the present note, we discuss the connection of the convex geometric and the analytic properties of the underlying manifold. As an application of the main results we show that in Riemannian manifolds of dimension at least three, local versions the Peano property, the drop completeness, or the “thinness” of triangles are equivalent with having constant curvature, and global versions characterize Euclidean and hyperbolic spaces among Cartan–Hadamard manifolds.

An Inertial Subgradient Extragradient Method for Approximating Solutions to Equilibrium Problems in Hadamard Manifolds

In this work, we are concerned with the iterative approximation of solutions to equilibrium problems in the framework of Hadamard manifolds. We introduce a subgradient extragradient type method with a self-adaptive step size. The use of a step size which is allowed to increase per iteration is to avoid the dependence of our method on the Lipschitz constant of the underlying operator as has been the case in recent articles in this direction. In general, operators satisfying weak monotonicity conditions seem to be more applicable in practice. By using inertial and viscosity techniques, we establish a convergence result for solving a pseudomonotone equilibrium problem under some appropriate conditions. As applications, we use our method to solve some theoretical optimization problems. Finally, we present some numerical illustrations in order to demonstrate the quantitative efficacy and superiority of our proposed method over a previous method present in the literature.

Translating Solitons Over Cartan–Hadamard Manifolds

We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan–Hadamard manifolds. We show that the asymptotic behavior of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions.

Implicit viscosity iterative algorithm for nonexpansive mapping on Hadamard manifolds

Implicit viscosity iterative algorithm for nonexpansive mapping on Hadamard manifolds

Compact Minimal Submanifolds in a Large Class of Riemannian Manifolds

Through a new technique, we provide uniqueness, rigidity and non-existence results for compact minimal submanifolds of arbitrary dimension in a large class of Riemannian manifolds, which include between others, Riemannian double-twisted and warped products. Moreover, we show that our results can be applied in particular, to space forms and Cartan–Hadamard manifolds, re-obtaining several classic results in a different approach. Interesting applications to Geometric Analysis are also showed.