Abstract

Let \(X\) and \(Y\) be Banach spaces. Let \(f:\Omega\to Y\) be a Fréchet differentiable function on an open subset \(\Omega\) of \(X\) and \(F\) be a set-valued mapping with closed graph. Consider the following generalized equation problem: \(0\in f(x)+F(x)\). In the present paper, we study a variant of Newton’s method for solving generalized equation and analyze semilocal and local convergence of the this method under weaker conditions than those associated by Jean-Alexis and Piétrus [13]. In fact, we show that the variant of Newton’s method is superlinearly convergent when the Fréchet derivative of \(f\) is \((L,p)\)-Hölder continuous and \((f+F)^{-1}\) is Lipzchitz-like at a reference point. Moreover, applications of this method to a nonlinear programming problem and a variational inequality are given. Numerical experiments are presented, which illustrate the theoretical results.

中文翻译：

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Lipschitz-Like 映射及其在牛顿法变体收敛分析中的应用

摘要

设\(X\)和\(Y\)是 Banach 空间。让\（F：\欧米茄\到Y \）是在一个开放的子集的Fréchet可微函数\（\欧米茄\）的\（X \）和\（F \）是具有封闭图形的一组值的映射。考虑以下广义方程问题：\(0\in f(x)+F(x)\)。在本文中，我们研究了求解广义方程的牛顿方法的变体，并在比 Jean-Alexis 和 Piétrus [13] 相关的条件更弱的条件下分析了该方法的半局部和局部收敛性。事实上，我们证明当\(f\)的 Fréchet 导数为\((L,p)\)时，牛顿方法的变体是超线性收敛的-Hölder 连续且\((f+F)^{-1}\)在参考点处类似于 Lipzchitz。此外，还给出了该方法在非线性规划问题和变分不等式中的应用。给出了数值实验，说明了理论结果。