## Uniform convergence of the Newton method for Aubin continuous maps.(English)Zbl 0865.90115

Summary: We prove that the Newton method applied to the generalized equation $$y\in f(x)+F(x)$$ with a $$C^1$$ function $$f$$ and a set-valued map $$F$$ acting in Banach spaces, is locally convergent uniformly in the parameter $$y$$ if and only if the map $$(f+F)^{-1}$$ is Aubin continuous at the reference point. We also show that the Aubin continuity actually implies uniform $$Q$$-quadratic convergence provided that the derivative of $$f$$ is Lipschitz continuous. As an application, we give a characterization of the uniform local $$Q$$-quadratic convergence of the sequential quadratic programming method applied to a perturbed nonlinear program.

### MSC:

 90C30 Nonlinear programming 49M37 Numerical methods based on nonlinear programming 65K10 Numerical optimization and variational techniques 47H04 Set-valued operators
Full Text: