Positive quadratic minimization under one affine constraint (${\min }_{a^{T}z=b}{z}^{T}Hz=b^2/(a^T{H}^{-1}a)$)

Let $H$ be a symmetric positive-definite matrix, $a\ne 0$, and $b\in \mathbb{R}$. Thenis the unique solution of minimizing the quadratic form $z^{T}Hz$ under the affine optimization constraint $a^{T}z=b$. The minimum value is $b^2/(a^T{H}^{-1}a)$.

The denominator is positive because $H^{-1}$ is a positive-definite matrix. The Lagrange multiplier equations $2Hz=\lambda a$ give the displayed candidate and verify the constraint. For a global proof, any feasible perturbation $u=z-z_{\ast }$ satisfies $a^{T}u=0$. HenceThis is positive for every nonzero $u$. The result also handles several independent quadratic blocks by taking $H$ to be their block diagonal matrix and concatenating their variables.