Convergence Analysis of an Inexact Gradient Method for Smooth Optimization

Gradient-based algorithms have been employed to solve numerous problems due to their efficiency, particularly in large-scale scenarios. The convergence theories of these algorithms rely on the assumption that exact first-order information is available. However, these algorithms frequently operate with inexact information regarding the optimization problem, such as the objective function and its gradient. In recent decades, studies have analyzed the convergence of these algorithms in inexact environments for solving both optimization problems and systems of nonlinear equations. This study presents a convergence analysis of an inexact gradient method for smooth and unconstrained optimization, considering three distinct step size strategies: constant, variable, and diminishing. The analysis demonstrates that the method converges to stationary points under the condition that the inexact gradient error is upper bounded by a quadratically summable sequence of error tolerances. The results obtained from a numerical experiment strongly support the proposed convergence theory.