Random and cyclic projection algorithms for variational inequalities

ABSTRACT

In this paper, we propose a new incremental constraint projection method (containing random projection method and cyclic projection method) for solving variational inequality problems in $R^n$, where the underlying function is Lipschitz continuous and monotone plus. We focus on special structures that lend themselves to sampling, such as when X is the intersection of a large number of sets, and/or F is an expected value or is the sum of a large number of component functions. Our method requires only two projections onto a suitable halfspace which replaces the projections onto constrained set $X_k$. We prove the sequence generated by our method is globally convergent to a solution of the variational inequalities in almost sure sense both random projection method and cyclic projection method. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.

Keywords:

• Random variables
• random projection method
• monotone plus mapping
• variational inequality

2010 Mathematics Subject Classification:

• 58E35

Acknowledgements

The authors would like to thanks the editor in chief and the anonymous referees for their valuable remarks and suggestions which helped to improve the original version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 11701050].