Journal
SIAM JOURNAL ON OPTIMIZATION
Volume 22, Issue 3, Pages 914-935Publisher
SIAM PUBLICATIONS
DOI: 10.1137/11083085X
Keywords
complexity; extragradient; variational inequality; maximal monotone operator; proximal point; ergodic convergence; hybrid; Newton methods
Categories
Funding
- NSF [CCF-0808863, CMMI-0900094]
- ONR [ONR N00014-11-1-0062]
- CNPq [474944/2010-7, 303583/2008-8]
- FAPERJ [E-26/110.821/2008]
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In a recent paper by Monteiro and Svaiter, a hybrid proximal extragradient (HPE) framework was used to study the iteration-complexity of a first-order (or, in the context of optimization, second-order) method for solving monotone nonlinear equations. The purpose of this paper is to extend this analysis to study a prox-type first-order method for monotone smooth variational inequalities and inclusion problems consisting of the sum of a smooth monotone map and a maximal monotone point-to-set operator. Each iteration of the method computes an approximate solution of a proximal subproblem obtained by linearizing the smooth part of the operator in the corresponding proximal equation for the original problem, which is then used to perform an extragradient step as prescribed by the HPE framework. Both pointwise and ergodic iteration-complexity results are derived for the aforementioned first-order method using corresponding results obtained here for a subfamily of the HPE framework.
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