An inexact proximal point algorithm for maximal monotone vector fields on Hadamard manifolds

2013 ◽  
Vol 41 (6) ◽  
pp. 586-591 ◽  
Author(s):  
Guo-ji Tang ◽  
Nan-jing Huang
Keyword(s):  
Proximal Point Algorithm ◽  
Vector Fields ◽  
Maximal Monotone ◽  
Hadamard Manifolds ◽  
Proximal Point ◽  
Monotone Vector Fields ◽  
Inexact Proximal ◽  
Maximal Monotone Vector Fields

scholarly journals An extragradient-type algorithm for variational inequality on Hadamard manifolds

2020 ◽  
Vol 26 ◽  
pp. 63 ◽  
Author(s):  
E.E.A. Batista ◽  
G.C. Bento ◽  
O.P. Ferreira
Keyword(s):  
Variational Inequality ◽  
Vector Field ◽  
Vector Fields ◽  
Extragradient Method ◽  
Maximal Monotone ◽  
Hadamard Manifolds ◽  
Convergence Properties ◽  
Type Algorithm ◽  
Monotone Vector Fields ◽  
Maximal Monotone Vector Fields

This paper presents an extragradient method for variational inequality associated with a point-to-set vector field in Hadamard manifolds, and a study of its convergence properties. To present our method, the concept of ϵ-enlargement of maximal monotone vector fields is used, and its lower-semicontinuity is established to obtain the method convergence in this new context.

scholarly journals Homogenization of Maximal Monotone Vector Fields via Selfdual Variational Calculus

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Nassif Ghoussoub ◽  
Abbas Moameni ◽  
Ramón Zárate Sáiz
Keyword(s):  
Vector Fields ◽  
Maximal Monotone ◽  
Variational Calculus ◽  
Classical Case ◽  
Divergence Form ◽  
Graph Convergence ◽  
Monotone Vector Fields ◽  
Γ Convergence ◽  
Maximal Monotone Vector Fields ◽  
Periodic Family

AbstractWe use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using Γ-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.

Inexact Proximal Point Methods for Multiobjective Quasiconvex Minimization on Hadamard Manifolds

2020 ◽  
Vol 186 (3) ◽  
pp. 879-898
Author(s):  
Erik Alex Papa Quiroz ◽  
Nancy Baygorrea Cusihuallpa ◽  
Nelson Maculan
Keyword(s):  
Hadamard Manifolds ◽  
Proximal Point ◽  
Proximal Point Methods ◽  
Inexact Proximal

An inexact algorithm with proximal distances for variational inequalities

2018 ◽  
Vol 52 (1) ◽  
pp. 159-176 ◽  
Author(s):  
E.A. Papa Quiroz ◽  
L. Mallma Ramirez ◽  
P.R. Oliveira
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Proximal Point Algorithm ◽  
Convergence Properties ◽  
Proximal Point ◽  
Extra Condition ◽  
Bregman Distances ◽  
Optim Theory ◽  
Inexact Proximal ◽  
Inexact Algorithm

In this paper we introduce an inexact proximal point algorithm using proximal distances for solving variational inequality problems when the mapping is pseudomonotone or quasimonotone. Under some natural assumptions we prove that the sequence generated by the algorithm is convergent for the pseudomonotone case and assuming an extra condition on the solution set we prove the convergence for the quasimonotone case. This approach unifies the results obtained by Auslender et al. [Math Oper. Res. 24 (1999) 644–688] and Brito et al. [J. Optim. Theory Appl. 154 (2012) 217–234] and extends the convergence properties for the class of φ-divergence distances and Bregman distances.

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