Efficient Douglas–Rachford Methods on Hadamard Manifolds with Applications to the Heron Problems

We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation--like regularizing term. To solve the convex minimization problem, we extend the Douglas--Rachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the Douglas--Rachford algorithm is reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for Krasnoselski--Mann iterations imply the convergence of the Douglas--Rachford algorithm. Unfortunately, these general results do not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions---namely, several distance-like functions and an indicator function of a special convex set. We prove that the reflections of certain distance-like functions on Hadamard manifolds are nonexpansive, which is an interesting result on its own. Furthermore, the reflection of the involved indicator function is nonexpansive on Hadamard manifolds with constant curvature so that the Douglas--Rachford algorithm converges here. Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods such as the cyclic proximal point algorithm and half-quadratic minimization. Numerical convergence is also observed in our experiments on the Hadamard manifold of symmetric positive definite matrices with the affine invariant metric, which does not have a constant curvature.

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.

The proximal point algorithm has many interesting applications, such as signal recovery, signal processing and others. In recent years, the proximal point method has been extended to Riemannian manifolds. The main advantages of these extensions are that nonconvex problems in classic sense may become geodesic convex by introducing an appropriate Riemannian metric, constrained optimization problems may be seen as unconstrained ones. In this paper, we propose an inexact proximal point algorithm for geodesic convex vector function on Hadamard manifolds. Under the assumption that the objective function is coercive, the sequence generated by this algorithm converges to a Pareto critical point. When the objective function is coercive and strictly geodesic convex, the sequence generated by this algorithm converges to a Pareto optimal point. Furthermore, under the weaker growth condition, we prove that the inexact proximal point algorithm has linear/superlinear convergence rate.

This article deals with the classes of approximate Minty- and Stampacchia-type vector variational inequalities on Hadamard manifolds and a class of nonsmooth interval-valued vector optimization problems. By using the Clarke subdifferentials, we define a new class of functions on Hadamard manifolds, namely, the geodesic LU-approximately convex functions. Under geodesic LU-approximate convexity hypothesis, we derive the relationship between the solutions of these approximate vector variational inequalities and nonsmooth interval-valued vector optimization problems. This paper extends and generalizes some existing results in the literature.

This article has two objectives. Firstly, we use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of generalized approximate geodesic convex functions and illustrated them with examples. We see the minimum requirements under which critical points, solutions of Stampacchia, and Minty weak variational-like inequalities and local approximate weakly efficient solutions can be identified, extending previous results from the literature for linear Euclidean spaces. Secondly, we show an economical application, again using solutions of the variational problems to identify Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions.

In this paper, a new class of roughly geodesic $B$-invex sets, quasi roughly geodesic $B$-invex functions and pseudo roughly geodesic $B$-invex functions are introduced and studied on Hadamard manifolds by relaxing the definitions of geodesic convex sets and functions. Some properties of quasi roughly geodesic $B$-invex functions and pseudo roughly geodesic $B$-invex functions are proved on Hadamard manifolds. As applications, some sufficient and necessary conditions for optimal solution of the nonlinear programming problems involving the quasi roughly geodesic $B$-invex functions and the pseudo roughly geodesic $B$-invex functions are given on Hadamard manifolds. The Mond-weir type dual problems for the nonlinear programming problems are also considered on Hadamard manifolds.

This article aims to introduce and analyze the viscosity method for hierarchical variational inequalities involving a ϕ-contraction mapping defined over a common solution set of variational inclusion and fixed points of a nonexpansive mapping on Hadamard manifolds. Several consequences of the composed method and its convergence theorem are presented. The convergence results of this article generalize and extend some existing results from Hilbert/Banach spaces and from Hadamard manifolds. We also present an application to a nonsmooth optimization problem. Finally, we clarify the convergence analysis of the proposed method by some computational numerical experiments in Hadamard manifold.

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.

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.

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.

In this paper, we point out that a recent characterization of geodesic convex hull on Hadamard manifolds is not rigorous and explain why the characterization does not hold like it in linear spaces. Therefore, a definition of geodesic pseudo-convex combination is proposed to show that the Knaster–Kuratowski–Mazurkiewicz theorem still holds under some mild conditions on Hadamard manifolds.

Variational inequalities on Hadamard manifolds

In this paper, as applications of an open-valued KKM-theorem, we prove an analogous KKM-theorem for open geodesic convex valued multimaps, and next give a Browder-type xed point theorem in a Hadamard manifold. Finally, we will give an existence theorem of maximal element in Hadamard manifolds. Mathematics Subject Classication: 58C30, 49J53

The purpose of this paper is to introduce a new splitting iterative algorithm for finding a common solution of a system of quasi-variational inclusion problems, equilibrium problems and fixed point for quasi-nonexpansive mappings in Hadamard manifolds. It is shown that under mild conditions, the iterative sequence generated by the proposed algorithm converges to a common solution. As applications, we apply our results to study the minimization problems and saddle point problems in Hadamard manifolds.

The present work is motivated by the problem of Bayesian inference for Gaussian distributions in symmetric Hadamard spaces (that is, Hadamard manifolds which are also symmetric spaces). To investigate this problem, it introduces new tools for Markov Chain Monte Carlo, and convex optimisation: (1) it provides easy-to-verify sufficient conditions for the geometric ergodicity of an isotropic Metropolis-Hastings Markov chain, in a symmetric Hadamard space, (2) it shows how the Riemannian gradient descent method can achieve an exponential rate of convergence, when applied to a strongly convex function, on a Hadamard manifold. Using these tools, two Bayesian estimators, maximum-a-posteriori and minimum-mean-squares, are compared. When the underlying Hadamard manifold is a space of constant negative curvature, they are found to be surprisingly close to each other. This leads to an open problem: are these two estimators, in fact, equal (assuming constant negative curvature)?KeywordsHadamard manifoldGaussian distributionBayesian inferenceMCMCConvex optimisation