An extrapolated projection and contraction algorithm with past iterates

This paper introduces a novel projection and contraction algorithm enhanced with past extrapolation for solving variational inequality problems in real Hilbert spaces. The key innovation lies in incorporating extrapolation from previous iterates, which reduces the computational cost from two operator evaluations per iteration in the original projection and contraction algorithm to only one evaluation. Under standard assumptions of pseudo-monotonicity and Lipschitz continuity, we establish the weak convergence of the generated sequence to a solution of the variational inequality. Furthermore, we derive non-asymptotic error bounds for the ergodic iterates via the gap function, proving a convergence rate of. Numerical experiments demonstrate the superior efficiency of our method compared to existing related algorithms in the literature.