I.6.3: Rate of Convergenc

Because of the structure of $f$ we know that $\mathcal{D}f(x)= \mathcal{D}_1g(x,y(x))$, where $y(x)$ is defined implicitly by $f(x)=g(x,y(x))$, or more explicitly by $$y(x)=\mathop{\mathbf{argmin}}\limits_{y\in\mathcal{Y}} g(x,y),$$ or, in the unconstrained and differentiable case, $$\mathcal{D}_2g(x,y(x))=0.$$ Differentiating again gives $$\mathcal{D}^2f(x)=\mathcal{D}_{11}g(x,y(x))-\mathcal{D}_{12}g(x,y(x)) [\mathcal{D}_{22}g(x,y(x))]^{-1}\mathcal{D}_{21}g(x,y(x)),$$

It follows that $$\mathcal{M}(x)=\mathcal{I}-[\mathcal{D}_{11}g(x,y(x))]^{-1}\mathcal{D}^2f(x).$$ This shows how the iteration matrix does not depend (directly) on the derivatives of $g$ with respect to $y,$ and can be interpreted as one minus the curvature of the function at the minimum, relative to the curvature of the augmentation function.