New extrapolation projection contraction algorithms based on the golden ratio for pseudo-monotone variational inequalities

Abstract

In real Hilbert spaces, for the purpose of trying to deal with the pseudo-monotone variational inequalities problem, we present a new extrapolation projection contraction algorithm based on the golden ratio in this study. Unlike ordinary inertial extrapolation, the algorithms are constructed based on a convex combined structure about the entire iterative trajectory. Extrapolation parameter $ \psi $ is selected in a more relaxed range instead of only taking the golden ratio $ \phi = \frac{\sqrt{5}+1 }{2} $ as the upper bound. Second, we propose an alternating extrapolation projection contraction algorithm to better increase the convergence effects of the extrapolation projection contraction algorithm based on the golden ratio. All our algorithms employ non-constantly decreasing adaptive step-sizes. The weak convergence results of the two algorithms are established for the pseudo-monotone variational inequalities. Additionally, the R-linear convergence results are investigated for strongly pseudo-monotone variational inequalities. Finally, we show the validity and superiority of the suggested methods with several numerical experiments. The numerical results show that alternating extrapolation does have obvious acceleration effect in practical application compared with no alternating extrapolation. Thus, the obvious effect of relaxing the selection range of parameter $ \psi $ on our two algorithms is clearly demonstrated.

Keywords

• golden ratio
• variational inequalities
• linear rate
• pseudo-monotone operator
• projection algorithms
• weak convergence