Variational Inequalities for Set-Valued Vector Fields on Riemannian Manifolds: Convexity of the Solution Set and the Proximal Point Algorithm

2012 ◽  
Vol 50 (4) ◽  
pp. 2486-2514 ◽  
Author(s):  
Chong Li ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequalities ◽  
Riemannian Manifolds ◽  
Proximal Point Algorithm ◽  
Vector Fields ◽  
Solution Set ◽  
Proximal Point

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.

Proximal Point Algorithm On Riemannian Manifolds

Optimization ◽  
2002 ◽  
Vol 51 (2) ◽  
pp. 257-270 ◽  
Author(s):  
O. P. Ferreira ◽  
P. R. Oliveira
Keyword(s):  
Riemannian Manifolds ◽  
Proximal Point Algorithm ◽  
Proximal Point
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