A new projection method for a class of variational inequalities

Abstract

In this paper, we revisit the numerical approach to variational inequality problems involving strongly monotone and Lipschitz continuous operators by a variant of projected reflected gradient method. Contrary to what done so far, the resulting algorithm uses a new simple stepsize sequence which is diminishing and nonsummable. This brings the main advantages of the algorithm where the construction of aproximation solutions and the formulation of convergence are done without the prior knowledge of the Lipschitz and strongly monotone constants of cost operators. The assumptions in the formulation of theorem of convergence are also discussed in this paper. Numerical results are reported to illustrate the behavior of the new algorithm and also to compare with others.

Keywords:

• Variational inequality
• strongly monotone operator
• Lipschitz continuity
• Projection method

AMS Subject Classifications:

• 65J15
• 65Y05
• 47H05
• 47J25
• 91B50

Acknowledgements

The authors would like to thank the Associate Editor and two anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper.

Notes

No potential conflict of interest was reported by the authors.