Extended Newton methods for conic inequalities: Approximate solutions and the extended Smale α-theory

Using the convex process theory and the majorizing technique, we study the convergence issues of the iterative sequences generated by the extended Newton method for solving the conic inequality system F≥C0F≥C0 defined by a cone C and a Fréchet differentiable function F satisfying the (extended) weak γ-condition. Convergence criterion of the extended Newton method is presented. As an application, we establish, when F is analytic, an extended α-theory similar to Smale's α-theory for nonlinear analytic equations.