Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 5 / b

Complete the square to write the objective asLet $z=(x,y)$ and $a=(1,2)$, so $\Vert a\Vert =\sqrt{5}$ while feasibility requires $\Vert z\Vert \ge \sqrt{20}=2\sqrt{5}$. The reverse triangle inequality givesThe objective is therefore at least zero. Equality is attained by $z=2a=(2,4)$, which lies on the boundary. Equality requires the vectors to have the same direction and $\Vert z\Vert =2\sqrt{5}$, so this minimizer is unique. HenceIt also satisfies the Karush-Kuhn-Tucker conditions with $\lambda =1/2$. The direct distance argument proves global optimality even though the exterior of a disk is not a convex set; this is nearest-point minimization outside a Euclidean ball.