scholarly journals On the generalized proximal point algorithm with applications to inclusion problems

2009 ◽  
Vol 5 (2) ◽  
pp. 381-390 ◽  
Author(s):  
Ram U. Verma ◽  
Keyword(s):  
Proximal Point Algorithm ◽  
Proximal Point ◽  
Inclusion Problems ◽  
Generalized Proximal Point Algorithm

scholarly journals Relatively Inexact Proximal Point Algorithm and Linear Convergence Analysis

2009 ◽  
Vol 2009 ◽  
pp. 1-11
Author(s):  
Ram U. Verma
Keyword(s):  
Convergence Analysis ◽  
Proximal Point Algorithm ◽  
Evolution Equations ◽  
Real Hilbert Space ◽  
Linear Convergence ◽  
Proximal Point ◽  
Evolution Inclusions ◽  
Inclusion Problems ◽  
Set Valued Mapping ◽  
Inexact Proximal

Based on a notion ofrelatively maximal(m)-relaxed monotonicity, the approximation solvability of a general class of inclusion problems is discussed, while generalizing Rockafellar's theorem (1976) on linear convergence using the proximal point algorithm in a real Hilbert space setting. Convergence analysis, based on this new model, is simpler and compact than that of the celebrated technique of Rockafellar in which the Lipschitz continuity at 0 of the inverse of the set-valued mapping is applied. Furthermore, it can be used to generalize the Yosida approximation, which, in turn, can be applied to first-order evolution equations as well as evolution inclusions.

scholarly journals An Approximate Proximal Point Algorithm for Maximal Monotone Inclusion Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Lingling Huang ◽  
Sanyang Liu ◽  
Weifeng Gao
Keyword(s):  
Hilbert Spaces ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Computational Results ◽  
Monotone Inclusion ◽  
Proximal Point ◽  
Inclusion Problems ◽  
Correction Step

This paper presents and analyzes a strongly convergent approximate proximal point algorithm for finding zeros of maximal monotone operators in Hilbert spaces. The proposed method combines the proximal subproblem with a more general correction step which takes advantage of more information on the existing iterations. As applications, convex programming problems and generalized variational inequalities are considered. Some preliminary computational results are reported.

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