Past exam of the continuous optimisation course of the University of Oxford / 2011 / 2 / ii

A standard form of the Zoutendijk theorem assumes that $f$ is continuously differentiable and bounded below on the starting level set, that its gradient is Lipschitz on a neighborhood of that set, and that the steps satisfy fixed Wolfe conditions. For descent directions $p_k$, defineThenIn particular, if $\cos {\theta }_k\ge \delta >0$, then $\nabla f(x_k)\to 0$. This last angle hypothesis is needed to turn the general descent theorem into gradient convergence.

For this quadratic, $f\ge 0$, the Hessian matrix is the constant positive-definite matrix $H$, and the gradient is globally Lipschitz with constant $L=2\max (c^{-2},d^{-2})$. Its level sets are compact. Thus all objective-function hypotheses hold; Wolfe holds for the exact steps with the parameters above. The direction hypothesis will be verified for alternating exact coordinate minimization in the next part.