Two self-adaptive stepsize relaxed dual inertial subgradient extragradient methods for solving pseudomonotone variational inequalities and demicontractive fixed point problems

Abstract

This paper introduces two new relaxed dual inertial extragradient-type methods with a self-adaptive stepsize rule for solving pseudomonotone variational inequality problems and fixed-point problems involving η-demicontractive mappings in real Hilbert spaces. The proposed self-adaptive stepsize rule eliminates the need for prior knowledge of the Lipschitz constant of the monotone operator. The iterative sequences exhibit strong convergence, and the dual-inertial approach notably accelerates the convergence speed. Unlike traditional methods, the proposed algorithms do not require line search procedures, as the stepsize is dynamically updated at each iteration based on preceding values. Convergence of the methods is well-established under mild assumptions. Numerical examples are presented to illustrate the effective implementation of our approach, demonstrating its efficiency, feasibility, and practical parameter selection. © The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional 2025.