"This set equals Tx only if x is feasible."
-->
"which is assumed nonempty for the current discussion. This set equals ..." (Note: change the period at the end of last equation to a comma).
"The distance between~$\mathcal{F}{\bar{\iter}}^{\mathsf{L}}$ and~$\mathcal{T}{\bar{\iter}}$ is ... as can be seen by considering the distance from~$0 \in \mathcal{T}{\bar{\iter}}$ to~$\mathcal{F}{\bar{\iter}}^{\mathsf{L}}$."
-->
The distance between~$\mathcal{F}{\bar{\iter}}^{\mathsf{L}}$ and~$\mathcal{T}{\bar{\iter}}$ is~$\delta{\bar{\iter}} = \norm{\step^}= \norm{\nabla h(\bar{\iter})^{\dagger} h(\bar{\iter})}$, where~$\step^$ is the projection of~$0\in \mathcal{T}{\bar{\iter}}$ onto~$\mathcal{F}_{\bar{\iter}}^{\mathsf{L}}$, and~$\nabla h(\bar{\iter})^{\dagger}$ denotes the Moore-Penrose pseudoinverse of~$\nabla h(\bar{\iter})$.
Indeed, if~$h$ is twice continuously differentiable, then its Taylor expansion provides~$\norm{h(\bar{\iter} + \step)} =\bigo(\norm{\step}^2)$ for all $\step$ in a bounded subset of~$\mathcal{F}{\bar{\iter}}^{\mathsf{L}}$, whereas~$\norm{h(\bar{\iter} + \step)} = \norm{h(\bar{\iter})} + \bigo(\norm{\step}^2)$ if~$\step$ stays in a bounded subset of~$\mathcal{T}{\bar{\iter}}$.
-->
Indeed, if~$h$ is twice continuously differentiable, then its Taylor expansion provides~$\norm{h(\bar{\iter} + \step)} =\bigo(\norm{\step}^2)$ for all $\step$ in a bounded subset of~$\mathcal{F}{\bar{\iter}}^{\mathsf{L}}$, which further implies~$\norm{h(\bar{\iter} + \step)} = \bigo(\norm{|h(\bar{\iter})|^2}) unless~$\norm{\step}$ is much larger than~$\norm{\step^*}$;
in contrast,~$\norm{h(\bar{\iter} + \step)} = \norm{h(\bar{\iter})} + \bigo(\norm{\step}^2)$ if~$\step$ stays in a bounded subset of~$\mathcal{T}{\bar{\iter}}$.
ragonneau
commented
2 years ago
https://github.com/ragonneau/phd-thesis/blob/9da0514a4cf45b0bd2bd4b68d0af06b05af2d56b/content/sqp.tex#L335
"This set equals Tx only if x is feasible." --> "which is assumed nonempty for the current discussion. This set equals ..." (Note: change the period at the end of last equation to a comma).
"The distance between~$\mathcal{F}{\bar{\iter}}^{\mathsf{L}}$ and~$\mathcal{T}{\bar{\iter}}$ is ... as can be seen by considering the distance from~$0 \in \mathcal{T}{\bar{\iter}}$ to~$\mathcal{F}{\bar{\iter}}^{\mathsf{L}}$." --> The distance between~$\mathcal{F}{\bar{\iter}}^{\mathsf{L}}$ and~$\mathcal{T}{\bar{\iter}}$ is~$\delta{\bar{\iter}} = \norm{\step^}= \norm{\nabla h(\bar{\iter})^{\dagger} h(\bar{\iter})}$, where~$\step^$ is the projection of~$0\in \mathcal{T}{\bar{\iter}}$ onto~$\mathcal{F}_{\bar{\iter}}^{\mathsf{L}}$, and~$\nabla h(\bar{\iter})^{\dagger}$ denotes the Moore-Penrose pseudoinverse of~$\nabla h(\bar{\iter})$.
Indeed, if~$h$ is twice continuously differentiable, then its Taylor expansion provides~$\norm{h(\bar{\iter} + \step)} =\bigo(\norm{\step}^2)$ for all $\step$ in a bounded subset of~$\mathcal{F}{\bar{\iter}}^{\mathsf{L}}$, whereas~$\norm{h(\bar{\iter} + \step)} = \norm{h(\bar{\iter})} + \bigo(\norm{\step}^2)$ if~$\step$ stays in a bounded subset of~$\mathcal{T}{\bar{\iter}}$. --> Indeed, if~$h$ is twice continuously differentiable, then its Taylor expansion provides~$\norm{h(\bar{\iter} + \step)} =\bigo(\norm{\step}^2)$ for all $\step$ in a bounded subset of~$\mathcal{F}{\bar{\iter}}^{\mathsf{L}}$, which further implies~$\norm{h(\bar{\iter} + \step)} = \bigo(\norm{|h(\bar{\iter})|^2}) unless~$\norm{\step}$ is much larger than~$\norm{\step^*}$; in contrast,~$\norm{h(\bar{\iter} + \step)} = \norm{h(\bar{\iter})} + \bigo(\norm{\step}^2)$ if~$\step$ stays in a bounded subset of~$\mathcal{T}{\bar{\iter}}$.