Department of Applied Mathematics & Physics, Kyoto Univiversity

Technical Report #98010 (April, 1998)

A Derivative-Free Line Search and DFP Method for Symmetric Equations with Global and Superlinear Convergence
by Donghui Li and Masao Fukushima

In this paper, we propose a derivative-free line search suited to iterative methods for solving systems of nonlinear equations with symmetric Jacobian matrices. The proposed line search can be implemented conveniently by a backtracking process and has such an attractive property that any iterative method with this line search generates a sequence of iterates that is approximately norm descent. Moreover, if the Jacobian matrices are uniformly nonsingular, then the generated sequence converges to the unique solution. We incorporate this line search with a Gauss-Newton based DFP method for solving symmetric equations. Under appropriate conditions, we establish global and superlinear convergence of the proposed DFP method. The obtained results show, in particular, that the proposed DFP method with inexact line search converges globally and superlinearly even for nonconvex unconstrained optimization problems and equality constrained optimization problems.