A Self-Adaptive Inertial Multi-Step Projection Method for Large-Scale Pseudomonotone Nonlinear Systems

Abstract

In this paper, we introduce an adaptive inertial extragradient method for approximating a common solution of a pseudomonotone equilibrium problem and a fixed-point problem associated with a mapping satisfying condition \((C_\mu)\) in a uniformly convex hyperbolic space. The proposed method integrates inertial extrapolation with an adaptive stepsize strategy within an extragradient framework. A key feature of the method is that the stepsize is updated without requiring prior knowledge of the Lipschitz-type constants of the equilibrium bifunction. Under appropriate assumptions on the bifunction, the mapping, and the control parameters, we establish the boundedness and asymptotic regularity of the generated sequence and prove its strong convergence to a common solution of the equilibrium and fixed-point problems without imposing compactness on the feasible set. Numerical experiments are provided to evaluate the computational performance of the proposed method and to compare it with related extragradient-type algorithms. The numerical results demonstrate the effectiveness and competitiveness of combining inertial extrapolation with an adaptive stepsize strategy in the nonlinear setting of uniformly convex hyperbolic spaces.