Sphere of Convergence of Newton's Method on Two Equivalent Systems from Nonlinear Programming

We study a local feature of two interior-point methods: a logarithmic barrier function method and a primal-dual method. In particular, we provide an asymptotic analysis on the radius of the sphere of convergence of Newton's method on two equivalent systems associated with the two aforementioned interior-point methods for nondegenerate nonlinear programs. We show that the radii of the spheres of convergence have different asymptotic behavior, as the two methods attempt to follow a solution trajectory {x } that, under suitable conditions, converges to a solution as 0. We show that, in the case of the barrier function method, the radius of the sphere of convergence of Newton's method is (), while for the primal-dual method the radius is bounded away from zero as 0. This work is an extension of the authors earlier work (Ref. 1) on linear programs.