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

Separate inactive and active optimization constraints. At an interior feasible point with $h>0$, a differentiable local minimum must satisfy $\nabla f=0$. Retain only solutions lying in the interior.

On the boundary $h=0$, if $\nabla h\ne 0$, apply the Lagrange multiplier condition. Using the optimization Lagrangian $L=f-\lambda h$, the Karush-Kuhn-Tucker conditions areThe sign follows from feasible motion into $h>0$: the directional derivative along $\nabla h$ at a minimum cannot be negative. Thus a boundary candidate has $h=0$ and a nonnegative Lagrange multiplier. Boundary points with $\nabla h=0$ must be examined separately, because the usual constraint qualification fails there.

Solve the interior and boundary systems, check membership in the feasible set, and establish which candidates are minima using second-order tests or a direct global comparison. The Karush-Kuhn-Tucker conditions alone are not sufficient for general nonconvex functions or feasible sets. On an unbounded feasible set, also check whether an infimum is attained and whether escaping sequences can improve the objective.