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On the Convergence Properties of Non-Euclidean Extragradient Methods for Variational Inequalities with Generalized Monotone Operators

Cong D. Dang (congdd***at***ufl.edu)
Guanghui Lan (glan***at***ise.ufl.edu)

Abstract: In this paper, we study a class of generalized monotone variational inequality (GMVI) problems whose operators are not necessarily monotone (e.g., pseudo-monotone). We present non-Euclidean extragradient (N-EG) methods for computing an approximate strong solution of these problems, and demonstrate how their iteration complexities depend on the global Lipschitz or H\"{o}lder continuity properties for their operators and the smoothness properties for the distance generating function used in the N-EG algorithms. We also introduce a variant of this algorithm by incorporating a simple line-search procedure to deal with problems with more general continuous operators. Numerical studies are conducted to illustrate the significant advantages of the developed algorithms over the existing ones for solving large-scale GMVI problems.

Keywords: Complexity, Monotone variational inequality, Pseudo-monotone variational inequality, Extragradient methods, Non-Euclidean methods, Prox-mapping

Category 1: Convex and Nonsmooth Optimization

Category 2: Complementarity and Variational Inequalities

Category 3: Nonlinear Optimization

Citation: Technical Report, Department of Industrial and Systems Engineering, University of Florida.

Download: [PDF]

Entry Submitted: 04/17/2012
Entry Accepted: 04/18/2012
Entry Last Modified: 11/11/2013

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