The Forward-Backward-Forward Method from discrete and continuous perspective for pseudo-monotone variational inequalities in Hilbert Spaces

Tseng's forward-backward-forward algorithm is a valuable alternative for Korpelevich's extragradient method when solving variational inequalities over a convex and closed set governed by monotone and Lipschitz continuous operators, as it requires in every step only one projection operation. However, it is well-known that Korpelevich's method is provable convergent and thus applicable when solving variational inequalities governed by a pseudo-monotone and Lipschitz continuous operator. In this paper we prove that Tseng's method converges also when it is applied to the solving of pseudo-monotone variational inequalities. In addition, we show that linear convergence is guaranteed under strong pseudo-monotonicity. We also associate a dynamical system to the pseudo-monotone variational inequality and carry out an asymptotic analysis for the generated trajectories. Numerical experiments show that Tseng's method outperforms Korplelevich's extragradient method when applied to the solving of pseudo-monotone variational inequalities and fractional programming problems.