Extended Newton-type Method for Nonsmooth Generalized Equation under n , α -point-based Approximation

Abstract

Let $\mathcal{X}$ and $\mathcal{Y}$ be Banach spaces and $\mathcal{\Omega }\subseteq X$ . Let $f:\mathcal{\Omega }⟶\mathcal{Y}$ be a single valued function which is nonsmooth. Suppose that $F:\mathcal{X\rightrightarrows }{2}^{\mathcal{Y}}$ is a set-valued mapping which has closed graph. In the present paper, we study the extended Newton-type method for solving the nonsmooth generalized equation $0\in f\left(x\right)+F\left(x\right)$ and analyze its semilocal and local convergence under the conditions that ${\left(f+F\right)}^{-1}$ is Lipschitz-like and $f$ admits a certain type of approximation which generalizes the concept of point-based approximation so-called $\left(n,\alpha \right)$ -point-based approximation. Applications of $\left(n,\alpha \right)$ -point-based approximation are provided for smooth functions in the cases $n=1$ and $n=2$ as well as for normal maps. In particular, when $0<\alpha <1$ and the derivative of $f$ , denoted $\nabla f$ , is $\left(\ell ,\alpha \right)$ -Hölder continuous, we have shown that $f$ admits $\left(1,\alpha \right)$ -point-based approximation for $n=1$ while $f$ admits $\left(2,\alpha \right)$ -point-based approximation for $n=2$ , when $0<\alpha <1$ and the second derivative of $f$ , denoted ${\nabla }^{2}f$ , is $\left(K,\alpha \right)$ -Hölder. Moreover, we have constructed an $\left(n,\alpha \right)$ -point-based approximation for the normal maps $f_{\mathcal{C}}+F$ when $f$ has an $\left(n,\alpha \right)$ -point-based approximation. Finally, a numerical experiment is provided to validate the theoretical result of this study.