Product-type tangential cone condition

Let $X,Y$ be real Hilbert spaces, let $U\subseteq X$ be open, and let $F:U\to Y$ be continuously Fréchet differentiable, meaning that $F$ is Fréchet differentiable at each point of $U$ and $x\mapsto F^{\mathrm{\prime }}(x)$ is continuous in operator norm. For a subset $V\subseteq U$ and a constant $c>0$, the product-type tangential cone condition on $V$ with constant $c$ is the requirement that, for every $x,\tilde{x}\in V$,

$\mathrm{\parallel }F(x)-F(\tilde{x})-F^{\mathrm{\prime }}(x)(x-\tilde{x}){\mathrm{\parallel }}_Y\le c\mathrm{\parallel }x-\tilde{x}{\mathrm{\parallel }}_X\mathrm{\parallel }F(x)-F(\tilde{x}){\mathrm{\parallel }}_Y\mathrm{.}$

Nonlinear forward map between real Hilbert spaces

Fréchet derivative on an open real Banach domain

One-basepoint local tangential cone condition

Christian Clason, Regularization of Inverse Problems, Assumption 11.10 and equation (11.23)

This is an explicit two-point product-remainder hypothesis. The LM stopping proof below assumes this inequality directly; it does not derive it from differentiability or from the one-basepoint local tangential cone condition.