A modified forward–backward splitting algorithm for solving nonmonotone variational inequalities

In 2000, Tseng proposed the forward–backward splitting algorithm (FBSA) for solving pseudomonotone variational inequalities (VI). To relax the monotonicity of the underlying mapping, Liu and Yang proposed a self-adaptive FBSA (SFBSA) for solving quasimonotone VI whenever the underlying mapping is Lipschitz continuous. Under an extra condition about the solution set, the global weak convergence of SFBSA is established. In this paper, we combine FBSA with projection and contraction method for solving nonmonotone VI in Euclidean space. The next iterate point of this algorithm is generated by projecting the next iterate point of FBSA onto a half-space, which maximizing the distance from former finite half-spaces to the current iterate point. The global convergence of this algorithm only requires the underlying mapping is continuous and the solution set of its dual variational inequalities is nonempty (without needing extra condition in SFBSA). Numerical experiments show that the modified FBSA (MFBSA) has fewer total number of iterations than SFBSA, which allows MFBSA faster than SFBSA in some examples from the CPU time point of view.