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On the convergence properties of non-Euclidean extragradient methods for variational inequalities with generalized monotone operators

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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 approximate strong solutions of these problems, and demonstrate how their iteration complexities depend on the global Lipschitz or Hölder 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.

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Acknowledgments

The author of this paper was partially supported by NSF Grants CMMI-1000347 and DMS-1319050, ONR Grant N00014-13-1-0036, and NSF CAREER Award CMMI-1254446.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL , 32611, USA

    Cong D. Dang & Guanghui Lan

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  1. Cong D. Dang
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  2. Guanghui Lan
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Correspondence to Guanghui Lan.

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Dang, C.D., Lan, G. On the convergence properties of non-Euclidean extragradient methods for variational inequalities with generalized monotone operators. Comput Optim Appl 60, 277–310 (2015). https://doi.org/10.1007/s10589-014-9673-9

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