Improved robust semi-local convergence analysis of Newton’s method for cone inclusion problem in Banach spaces under restricted convergence domains and majorant conditions. (English) Zbl 1379.65031
The article deals with the nonlinear inclusion problem
\[
F(x) \in C,
\]
where \(F:\;D (\subset {\mathbb X}) \to {\mathbb Y}\) is a continuously Fréchet differentiable function between a reflexive Banach space \({\mathbb X}\) and an arbitrary Banach space \({\mathbb Y}\), \(D\) an open set, \(C \subset {\mathbb Y}\) a nonempty closed convex cone. Approximate solutions \(x_k\) to this inclusion are defined by the equations
\[
x_{k+1} \in x_k + \text{argmin}\, \{\|d\|:\;F(x_k) + F'(x_k)d \in C\}, \quad k = 0,1,2,\dots,
\]
with an initial \(x_0 = \overline{x}\); it is assumed that \(F\) satisfies Robinson’s condition at \(\overline{x}\). The authors construct some scalar majorant equations for the inclusion under consideration and prove estimates
\[
\|x_{k+1} - x_k\| \leq t_{k+1} - t_k, \quad \|x_{k+1} - x_k\| \leq \frac{t_{k+1} - t_k}{(t_k - t_{k-1})^2} \, \|x_k - x_{k-1}\|^2, \qquad k = 0,1,2,\dots,
\]
where \(t_k\), \(k = 0,1,2,\dots\), are Newton approximations for majorant equations. In the end of the article the authors present a numerical example.
There are some vague places in the article; in particular, the Robinson condition is not defined in a correct way.
There are some vague places in the article; in particular, the Robinson condition is not defined in a correct way.
Reviewer: Peter P. Zabreĭko (Minsk)
MSC:
65J15 | Numerical solutions to equations with nonlinear operators |
49J52 | Nonsmooth analysis |
47J25 | Iterative procedures involving nonlinear operators |
47J22 | Variational and other types of inclusions |