Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space

Abstract

We present two extensions of Korpelevich's extragradient method for solving the variational inequality problem (VIP) in Euclidean space. In the first extension, we replace the second orthogonal projection onto the feasible set of the VIP in Korpelevich's extragradient method with a specific subgradient projection. The second extension allows projections onto the members of an infinite sequence of subsets which epi-converges to the feasible set of the VIP. We show that in both extensions the convergence of the method is preserved and present directions for further research.

Keywords:

• epi-convergence
• extragradient method
• Lipschitz mapping
• subgradient
• variational inequality

Acknowledgements

This work was partially supported by the United States-Israel Binational Science Foundation (BSF) Grant number 200912, and by Award Number R01HL070472 from the National Heart, Lung and Blood Institute. The third author was partially supported by the Israel Science Foundation (Grant 647/07), by the Fund for the Promotion of Research at the Technion and by the Technion President's Research Fund.