Multi-step-length gradient iterative method for separable nonlinear least squares problems

Separable nonlinear least squares (SNLLS) problems are critical in various research and application fields, such as image restoration, machine learning, and system identification.Solving such problems presents a challenge due to their nonlinearity.The traditional gradient iterative algorithm often zigzags towards the optimal solution and is sensitive to the initial guesses of unknown parameters.In this paper, we improve the convergence rate of the traditional gradient method by implementing a multi-step-length gradient iterative algorithm.Moreover, we incorporate the variable projection (VP) strategy, taking advantage of the separable structure observed in SNLLS problems.We propose a multi-step-length gradient iterative-based VP (Mul-GI-VP) method to solve such nonlinear optimization problems.Our simulation results verify the feasibility and high efficiency of the proposed algorithm.