Problem 3.5

The conclusion for this problem can be done without the preliminary problem and without the hint.

With the usual notation:

$$g=\nabla f(x_k), \quad B = B_k, \quad p=-B^{-1} g, \quad \vert\vert B^{-1}\vert\vert \vert\vert B\vert\vert \leq M;$$

and

$$y=B^{-\frac 12} g, \quad g=B^{\frac 12} y.$$

Then

$$\begin{eqnarray*} \cos \theta &=& \frac{g^t B^{-1} g}{\vert\vert B^{-1}g\vert\vert \vert\vert g\vert\vert} \\ \end{eqnarray*}$$

and

$$\begin{eqnarray*} \frac 1{\cos \theta} &=& \frac{\vert\vert B^{-1}g\vert\vert \... ...vert B^{\frac 12}\vert\vert \\ & \leq & M^{-\frac 12} \leq M. \end{eqnarray*}$$