On maximal monotonicity of bifunctions on Hadamard manifolds
type: Item
instance of: scholarly article
title: On maximal monotonicity of bifunctions on Hadamard manifolds
zbMATH Open document ID: 1476.90325
author: Pedro A. jun. Soares
author: J. X. da Cruz Neto
author: Flávia M. O. Jacinto
author: João Carlos O. Souza
published in: Journal of Global Optimization
publication date: 2018-12-07T00:00:00Z
review text: Assume that \(M\) is an \(n\)-dimensional Hadamard manifold. The \(n\)-dimensional tangent space of \(M\) at \(x,\) and the tangent bundle of \(M\)\ are, respectively, denoted by \(T_{x}M,\) and \(TM=\bigcup_{x\in M}T_{x}M\). Also, the geodesic \(\gamma=\gamma_{\nu}(\cdot,x)\) is determined by its position \(x\) and velocity \(\nu\) at \(x\). In addition, the exponential map \(\exp:T_{x}M\rightarrow M\) is defined by \(\exp_{x}\nu=\gamma_{\nu}(1,x)\), for each \(x\in M\). A set \(\Omega \subset M\) is called convex if \(\gamma:\left[ a,b\right] \rightarrow M\) is a geodesic such that \(x=\gamma (a)\in \Omega\) and \(y=\gamma (b)\in\Omega\), then \[ \gamma ((1-t)a+tb)\in \Omega, \ \ \ \forall t\in \left[ 0,1\right] . \] The convex hull of \(C\subset M\) is denoted by \(\operatorname{conv}(C)\). Let \(\Omega\subset M\) be a convex set. A function \(f:\Omega\rightarrow\mathbb{R}\) is called convex if for each geodesic segment \(\gamma:[a,b]\rightarrow\Omega\), with \(a<b\), the composition \(f\circ \gamma:[a,b]\rightarrow\mathbb{R}\) is convex. A vector \(s\in T_{p}M\) is said to be a subgradient of \(f\) at \(p\in\Omega\) if \[ f(q)\geq f(p)+\left \langle s,\exp_{p}^{-1}q\right \rangle ,\ \ \ \ \forall q\in \Omega. \] Let \(\chi (M)\) be the set of all set-valued vector fields \(A:M\rightarrow2^{TM}\) such that \(A(x)\subset T_{x}M\), for each \(x\in M\), and the domain \(\mathcal{D}(A)\) of \(A\) is closed and convex. A vector field \(T\in \chi (M)\) is said to be monotone if the following condition holds \[ \left \langle u,\exp_{x}^{-1}y\right \rangle \leq \left \langle v,-\exp_{y}^{-1}x\right \rangle ,\ \ \ \ \forall x,y\in \mathcal{D}(T) ,\ \forall u\in T(x)\ \text{ and }\ \forall v\in T(y). \] Also, \(T\in\) \(\chi (M)\) is called maximal monotone if it is monotone and the following implication holds for any \(x\in \mathcal{D}(T)\) and \(u\in T_{x}M\): \[ \left \langle u,\exp_{x}^{-1}y\right \rangle \leq \left \langle v,-\exp_{y}^{-1}x\right \rangle ,\ \ \ \ \for…
reviewed by: Mohammad Hossein Alizadeh
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 6990996
zbMATH Keywords: bifunction
zbMATH Keywords: monotone vector fields
zbMATH Keywords: equilibrium problems
zbMATH Keywords: Hadamard manifolds
MaRDI profile type: Publication
full work available at URL: s10898-018-0663-9
OpenAlex ID: W2805018461
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DOI: 10.1007/S10898-018-0663-9
description: scientific article; zbMATH DE number 6990996