  This paper delves into an in-depth exploration of the Variable Projection
(VP) algorithm, a powerful tool for solving separable nonlinear optimization
problems across multiple domains, including system identification, image
processing, and machine learning. We first establish a theoretical framework to
examine the effect of the approximate treatment of the coupling relationship
among parameters on the local convergence of the VP algorithm and theoretically
prove that the Kaufman's VP algorithm can achieve a similar convergence rate as
the Golub \& Pereyra's form. These studies fill the gap in the existing
convergence theory analysis, and provide a solid foundation for understanding
the mechanism of VP algorithm and broadening its application horizons.
Furthermore, drawing inspiration from these theoretical revelations, we design
a refined VP algorithm for handling separable nonlinear optimization problems
characterized by large residual, called VPLR, which boosts the convergence
performance by addressing the interdependence of parameters within the
separable model and by continually correcting the approximated Hessian matrix
to counteract the influence of large residual during the iterative process. The
effectiveness of this refined algorithm is corroborated through numerical
experimentation.
