• DOI:10.1137/14097238X
• Corpus ID: 18789486

Projected Reflected Gradient Methods for Monotone Variational Inequalities

@article{Malitsky2015ProjectedRG,
  title={Projected Reflected Gradient Methods for Monotone Variational Inequalities},
  author={Yura Malitsky},
  journal={SIAM J. Optim.},
  year={2015},
  volume={25},
  pages={502-520}
}

• Yura Malitsky
• Published 17 February 2015
• Mathematics, Computer Science
• SIAM J. Optim.

This paper is concerned with some new projection methods for solving variational inequality problems with monotone and Lipschitz-continuous 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 projection-type methods for variational inequalities with monotone mapping. Also we prove that it has R-linear 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 ve...