Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / B / 5 / iii / Solution

The global optimality conditions for a quadratic trust-region subproblem with Hessian $B$ are: for some $\lambda \ge 0$,Here $B=I$, so the shifted matrix is positive definite. If $\Vert g{\Vert }_2\le \Delta$, choose $\lambda =0$ and $s=-g$. If $\Vert g{\Vert }_2>\Delta$, the active boundary gives $s=-g/(1+\lambda )$ and $1+\lambda =\Vert g{\Vert }_2/\Delta$. HenceThis is exactly the Cauchy point. Because the model is strictly convex, it is the unique global minimizer, including $s_{\ast }=0$ when $g=0$.