Past exam of the continuous optimisation course of the University of Oxford / 2025 / 2 / a / i / Solution

The Cauchy point minimizes the model along the negative gradient ray inside the trust region. With $q=g^{T}Hg$, writeIf $q\le 0$, this last quadratic function decreases throughout the interval. If $q>0$, its unconstrained minimum occurs at $t=\Vert g{\Vert }^2/q$. ThusThe additive constant $a$ has no effect on the Cauchy point. To obtain the Cauchy decrease bound, first suppose $q>0$. Since $\tau q\le \Vert g{\Vert }^2$,The spectral norm inequality $q\le \Vert H\Vert \Vert g{\Vert }^2$ gives $\tau \ge \min \{\Delta /\Vert g\Vert ,1/\Vert H\Vert \}$. HenceIf $q\le 0$, the decrease is at least $\Delta \Vert g\Vert$, which also implies the displayed bound. When $H=0$, interpret $\Vert g\Vert /\Vert H\Vert =+\infty$. Thus the universal choice is $c=1/2$.