Weak convergence of an inertial projection-contraction algorithm with adaptive stepsize and golden-ratio relaxation for pseudomonotone variational inequalities

Abstract

This paper investigates an inertial projection-contraction algorithm for solving pseudomonotone variational inequality problems in real Hilbert spaces. The proposed method combines a modified Mann-type relaxation, inertial extrapolation, and an adaptive line-search mechanism that does not require prior knowledge of the Lipschitz constant of the underlying operator. Unlike standard projection-contraction schemes, the proposed algorithm is developed under pseudomonotonicity, uniform continuity, and weak-to-strong sequential continuity assumptions. A detailed weak convergence analysis is established using projection inequalities, asymptotic regularity arguments, and Opial's lemma. Furthermore, we discuss the role of the golden-ratio relaxation parameter in the stability of the proposed iteration. Numerical experiments involving finite-dimensional variational inequality problems and image deblurring applications are provided to illustrate the practical performance of the algorithm and to compare it with existing projection-contraction methods.