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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
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