Scalar multiplier certificate for a quadratic equality constraint ($\Vert c{\Vert }^2/(4{\lambda }^2)-\lambda /(2\gamma )=R$)

For $\gamma >0$ and $c\ne 0$, minimize $c^{T}u+\gamma z^2$ subject to $\Vert u{\Vert }^2+z=R$. 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.