Proposition: Optimal weights for a covered bank

Assume the complete evaluated ambiguity is $\mathrm{col}(M)$ and that the measurement covariance $\Sigma$ is positive definite. Set

and let $\Pi$ be the Euclidean orthogonal projection onto $\ker C^{\mathsf{T}}$. If $\Pi q\ne 0$, the unique weights minimizing $w^{\mathsf{T}}\Sigma w$ subject to

are

They are unchanged when $s$ is replaced by $s+M\theta$.

Proof. Put $y={\Sigma }^{1/2}w$. The constraints become $C^{\mathsf{T}}y=0$ and $q^{\mathsf{T}}y=1$, and the variance is ${\Vert y\Vert }^2$. On the permitted subspace, $q^{\mathsf{T}}y=(\Pi q{)}^{\mathsf{T}}y$. Cauchy–Schwarz gives the minimum at $y=\Pi q/{\Vert \Pi q\Vert }^2$. Finally, ${\Sigma }^{-1/2}M\theta \in \mathrm{col}(C)$, which $\Pi$ annihilates.