A Global Convergence Theory for Arbitrary Norm Trust-Region Methods for Nonlinear Equations

: In this work we extend the Levenberg-Marquardt algorithm for approximating zeros of the nonlinear system F(x) = 0, where F : IR (exp n) approaches the limit of IR (exp n) is continuously differentiable. Instead of the l2 norm, arbitrary norms can be used in the trust-region objective function and in the trust-region constraint. The algorithm is shown to be globally convergent. This research was motivated by the recent work of Duff, Nocedal and Reid. A key point in our analysis is that the tools from nonsmooth analysis and the Zangwill convergence theory allow us to establish essentially the same properties for an arbitrary norm trust-region algorithm that have been established for the Levenberg-Marquardt algorithm using the tools from smooth optimization. It is shown that all members of this class of algorithms locally reduce to Newton's method and that the iteration sequence actually converges to a solution.