Randomized iterative methods with Polyak step-size for solving generalized absolute value equations

Abstract: In this paper, we systematically incorporate the Polyak step-size into the randomized iterative method to improve its efficiency for solving generalized absolute value equations. In particular, we adopt the Polyak step-size within a stochastic iterative setting where the objective function updates dynamically at every step, unlike the classical Polyak step-size designed for deterministic optimization with fixed objective functions. Consequently, this novel implementation differs from the conventional Polyak scheme and demands a dedicated convergence analysis. We rigorously analyze the convergence properties of the proposed method and establish its linear convergence in expectation. Numerical experiments demonstrate that the incorporation of the Polyak step-size substantially improves the computational performance of randomized iterative methods with constant step-sizes.