On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators

Authors:
Xingju Cai;Guoyong Gu;Bingsheng He
Affiliations:
Department of Mathematics, Nanjing University, Nanjing, P.R. China 210093;Department of Mathematics, Nanjing University, Nanjing, P.R. China 210093;Department of Mathematics, Nanjing University, Nanjing, P.R. China 210093
Venue:
Computational Optimization and Applications
Year:
2014

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Abstract

Nemirovski's analysis (SIAM J. Optim. 15:229---251, 2005) indicates that the extragradient method has the O(1/t) convergence rate for variational inequalities with Lipschitz continuous monotone operators. For the same problems, in the last decades, a class of Fej茅r monotone projection and contraction methods is developed. Until now, only convergence results are available to these projection and contraction methods, though the numerical experiments indicate that they always outperform the extragradient method. The reason is that the former benefits from the `optimal' step size in the contraction sense. In this paper, we prove the convergence rate under a unified conceptual framework, which includes the projection and contraction methods as special cases and thus perfects the theory of the existing projection and contraction methods. Preliminary numerical results demonstrate that the projection and contraction methods converge twice faster than the extragradient method.