Projected-Reflected Subgradient-Extragradient Method and Its Real-World Applications

Abstract

Our main focus in this work is the classical variational inequality problem with Lipschitz continuous and pseudo-monotone mapping in real Hilbert spaces. An adaptive reflected subgradient-extragradient method is presented along with its weak convergence analysis. The novelty of the proposed method lies in the fact that only one projection onto the feasible set in each iteration is required, and there is no need to know/approximate the Lipschitz constant of the cost function a priori. To illustrate and emphasize the potential applicability of the new scheme, several numerical experiments and comparisons in tomography reconstruction, Nash–Cournot oligopolistic equilibrium, and more are presented.

Keywords

• subgradient-extragradient
• reflected step
• variational inequality
• pseudo-monotone mapping
• Lipschitz mapping