On projected alternating BB methods for variational inequalities

Abstract

The Barzilai–Borwein (BB) step size initially proposed in the context of unconstrained optimization has become one of the most popular step choices for gradient-based methods in the optimization community. It is well-known that the powerful variational inequality can be used to characterize the first-order optimality condition of constrained optimization problems. However, a variational inequality problem is not always equivalent to a constrained optimization problem. In this paper, we follow the spirit of BB step size and propose a projection method with alternate BB step size for general variational inequalities. Although the global convergence is established under some strong conditions, a series of computational experiments on nonlinear complementarity problems, image deblurring problems and generalized Nash equilibrium problems demonstrate that the proposed almost-parameter-free projection method is more efficient than some existing state-of-the-art projection methods in the literature.

Keywords:

• Projection method
• BB step size
• variational inequality problem
• Complementarity problem
• image deblurring

2010 Mathematics Subject Classifications:

• 65K05
• 90C30
• 90C33

Acknowledgements

The authors would like to thank the three anonymous referees for their close reading and valuable comments, which helped us improve the presentation of this paper.

Disclosure statement

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

Additional information

Funding

Y. Qu was supported by 2018 Zhejiang Provincial University Student Science and Technology Innovation Project (young talents plan) at grant number 2018R407035). H. He was supported in part by National Natural Science Foundation of China [grant numbers 11771113 and 11971138] and Natural Science Foundation of Zhejiang Province [grant number LY20A010018]. Y.-C. Liou was supported in part by the grant from Kaohsiung Medical University Research Foundation [grant number KMU-M108002], MOST 106-2923-E-039-001-MY3, and MOST 108-2410-H-037-020