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Yura Malitsky(y.malitskygmail.com) Abstract: This paper is concerned with some new projection methods for solving variational inequality problems with monotone and Lipschitzcontinuous mapping in Hilbert space. First, we propose the projected reflected gradient algorithm with a constant stepsize. It is similar to the projected gradient method, namely, the method requires only one projection onto the feasible set and only one value of the mapping per iteration. This distinguishes our method from most other projectiontype methods for variational inequalities with monotone mapping. Also we prove that it has Rlinear rate of convergence under the strong monotonicity assumption. The usual drawback of algorithms with constant stepsize is the requirement to know the Lipschitz constant of the mapping. To avoid this, we modify our first algorithm so that the algorithm needs at most two projections per iteration. In fact, our computational experience shows that such cases with two projections are very rare. This scheme, at least theoretically, seems to be very effective. All methods are shown to be globally convergent to a solution of the variational inequality. Preliminary results from numerical experiments are quite promising. Keywords: variational inequality, projection method, monotone mapping, extragradient method Category 1: Complementarity and Variational Inequalities Citation: SIAM J. Optim., 25(1), 2015, 502–520 Download: [PDF] Entry Submitted: 03/20/2015 Modify/Update this entry  
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