= Scalar multiplier certificate for a quadratic equality constraint
{title2=$\|c\|^2/(4\lambda^2)-\lambda/(2\gamma)=R$}
For $\gamma>0$ and $c\ne0$, minimize $c^Tu+\gamma z^2$ subject to $\|u\|^2+z=R$. Adding a positive <Lagrange multiplier> times the constraint gives a <positive-definite> quadratic <optimization Lagrangian>, whose minimizer is $u=-c/(2\lambda)$ and $z=-\lambda/(2\gamma)$. Feasibility is the displayed scalar equation. Its left side strictly decreases from positive infinity to negative infinity on $\lambda>0$, so it has a unique root for every real $R$. The <Lagrangian sufficiency theorem> certifies a unique global optimum on the possibly nonconvex equality surface.
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