Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings

• Published in: Optimization Vol. 70; no. 4; pp. 715 - 740
• Main Authors: Ceng, Lu-Chuan, Shang, Meijuan
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03.04.2021; Taylor & Francis LLC
• Subjects: Algorithms; Approximation; Asymptotic methods; Asymptotic properties; asymptotically nonexpansive mapping; common fixed point; Fixed points (mathematics); Hilbert space; Hybrid inertial subgradient extragradient method; Mapping; pseudomonotone variational inequality; Search process; sequentially weak continuity
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2019.1647203

In this paper, we introduce hybrid inertial subgradient extragradient algorithms with line-search process for solving a variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, we prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.