Local and superlinear convergence of structured quasi-newton methods for nonlinear optimization

Abstract

This paper is concerned with local and q-superlinear convergence of structured quasi-Newton methods for solving unconstrained and constrained optimization problems. These methods have been developed for solving optimization problems in which the Hessian matrix has a special structure. For example, Dennis, Gay and Welsch (1981) proposed the structured DFP update for nonlinear least squares problems and Tapia (1988) derived the structured BFGS update for equality constrained problems within the framework of the SQP method with the augmented Lagrangian function. Recently, Engels and Martinez (1991) unified these methods and showed local and q-superlinear convergence of the convex class of the structured Broyden family. In this paper, we extend the results of Engels and Martinez to a wider class of the structured Broyden family. We prove local and q-superlinear convergence of the method in a way different from the proof by Engels and Martinez. Our proof for convergence is based on the result by Stachurski (1981). Finally, we apply the convergence results to unconstrained nonlinear least squares problems and equality constrained minimization problems.