Is the solution to the following inequality-constrained least-norm problem unique?
$$\begin{array}{ll} \text{minimize} & \| r \|_2\\ \text{subject to} & Ar \leq b\\ & l \leq r \leq u\end{array}$$
Is the solution to least $2$-norm unique
convex optimizationoptimizationquadratic programming
Best Answer
The answer is yes. Suppose that $x\neq y$ are both minimizers with value $d^*$ (note that so long as the feasible region is nonempty, such minimizers exist by compactness). But then $\| \lambda x+(1-\lambda)y\|<d^*$ for $\lambda\in (0,1)$, as the unit ball is a strictly convex set. $\lambda x+(1-\lambda)y$ also satisfies the constraints as the feasible region is convex, so this gives a smaller value. Therefore, the minimizer must be unique.