Convergence of a quasi-Newton method for solving systems of nonlinear underdetermined equations

Abstract

The development and convergence analysis of a quasi-Newton method for the solution of systems of nonlinear underdetermined equations is investigated. These equations arise in many application fields, e.g., supervised learning of large overparameterised neural networks, which require the development of efficient methods with guaranteed convergence. In this paper, a new approach for the computation of the Moore–Penrose inverse of the approximate Jacobian coming from the Broyden update is presented and a semi-local convergence result for a damped quasi-Newton method is proved. The theoretical results are illustrated in detail for the case of systems of multidimensional quadratic equations, and validated in the context of eigenvalue problems and supervised learning of overparameterised neural networks.