Stability of Real Solutions to Nonlinear Equations and Its Applications

Pub Date : 2024-03-06 DOI:10.1134/s0081543823050012

A. V. Arutyunov, S. E. Zhukovskiy

引用次数: 0

Abstract

We study the stability of solutions to nonlinear equations in finite-dimensional spaces. Namely, we consider an equation of the form \(F(x)=\overline{y}\) in the neighborhood of a given solution \(\overline{x}\). For this equation we present sufficient conditions under which the equation \(F(x)+g(x)=y\) has a solution close to \(\overline{x}\) for all \(y\) close to \(\overline{y}\) and for all continuous perturbations \(g\) with sufficiently small uniform norm. The results are formulated in terms of \(\lambda\)-truncations and contain applications to necessary optimality conditions for a constrained optimization problem with equality-type constraints. We show that these results on \(\lambda\)-truncations are also meaningful in the case of degeneracy of the linear operator \(F'(\overline{x})\).

查看原文

非线性方程实解的稳定性及其应用

摘要我们研究有限维空间中非线性方程解的稳定性。也就是说，我们考虑在给定解 \(\overline{x}\)的邻域内形式为 \(F(x)=\overline{y}\)的方程。对于这个方程，我们提出了充分条件，在这些条件下，方程 \(F(x)+g(x)=y\) 对于所有接近 \(\overline{x}\) 的 \(y\) 和所有具有足够小的均匀规范的连续扰动 \(g\) 都有一个接近 \(\overline{x}\) 的解。这些结果是用\(\lambda\)-截断来表述的，并且包含了对具有相等类型约束的约束优化问题的必要最优条件的应用。我们证明了这些关于截断的结果在线性算子 \(F'(\overline{x})\)退化的情况下也是有意义的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文