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

Assume the denominators are defined, so $c,d\ne 0$. For a descent direction $p$, set $\phi (\alpha )=f(x+\alpha p)$. The Wolfe conditions areThe first is sufficient decrease and the second is the curvature condition. Hereis positive definite, with eigenvalues $2/c^2,2/d^2$. The exact step isIt gives ${\phi }^{\prime }({\alpha }_{\ast })=0$, so the curvature condition holds. Its decrease is $\phi ({\alpha }_{\ast })-\phi (0)={\alpha }_{\ast }{\phi }^{\prime }(0)/2$, so sufficient decrease holds for the usual choice $0<\sigma \le 1/2$, $\sigma <\eta <1$. Hence exact searches satisfy Wolfe with an Armijo constant at most one half. If one allowed $\sigma >1/2$, even an exact quadratic minimizer would fail that condition; the parameter range matters.