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Let $\|F'(\boldsymbol{\beta_1})-F'(\boldsymbol{\beta_2})\|_2 \leq M \| \boldsymbol{\beta_1}-\boldsymbol{\beta_2} \| _2,$ $\forall \boldsymbol{\beta_1},\boldsymbol{\beta_2} \in \overline{S(\boldsymbol{\beta}^*,\delta)},$ where \begin{equation}\overline{S(\boldsymbol{\beta}^*,\delta)}= \{ \boldsymbol{\beta} : \| \boldsymbol{\beta}^*-\boldsymbol{\beta} \|_2 \leq \delta \},\end{equation} and $\delta>0$. $\boldsymbol{\beta}^*$ is just the solution of the system $F(\boldsymbol{\beta})$ and $F'$ is its jacobian matrix. So $F'$ is Lipschitz. How can we prove that it holds:

$F(\boldsymbol{\beta})=F'(\boldsymbol{\beta})(\boldsymbol{\beta}-\boldsymbol{\beta}^*)+O(\|e\|_2^2)$ where $\|e\|=\boldsymbol{\beta}-\boldsymbol{\beta}^*$ ?

john
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  • the system is $F(\beta)=0$ of course... – john May 12 '15 at 18:52
  • Mean Value Theorem: $F(\beta)=F'(\gamma)(\beta-\beta^) =F'(\beta)(\beta-\beta^)+(F'(\gamma)-F'(\beta)(\beta-\beta^)$. Where $\gamma$ is between $\beta$ and $\beta^$. –  May 12 '15 at 18:52

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