Computational Optimization and Applications

, Volume 57, Issue 2, pp 339–363 | Cite as

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

  • Xingju Cai
  • Guoyong Gu
  • Bingsheng He


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 Fejér 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.


Variational inequality Projection and contraction method Convergence rate 



The authors thank X.-L. Fu, M. Li, M. Tao and X.-M. Yuan for the discussion and valuable suggestions.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingP.R. China

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