Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 339 / 2 / a / Solution

The objective is strictly convex, so the minimizer is unique. The Slater condition makes the Karush-Kuhn-Tucker conditions necessary and sufficient. Absorb the box constraints into the Euclidean projection onto a convex set and attach a scalar multiplier $\nu$ to $a^{T}x=b$. Stationarity over the box is equivalent towhile primal feasibility requires $a^T{x}^{\ast }=b$. Coordinatewise, these conditions areThey are also sufficient because they minimize the Lagrangian over the box and satisfy the equality constraint. Thus the projection onto a box-constrained hyperplane reduces to solving the displayed one-dimensional continuous, nonincreasing equation for $\nu$. The multiplier need not be unique on a flat interval, but the projected vector is unique.