How do I get only two constraints for this Lagrangian?

Question

In a paper I'm reading, they minimize a function $g(\alpha)$ subject to something like:

$a(\alpha)=b(\alpha)=c$ $\quad$ (Eq. 6 in the paper)

and

$d(\alpha)=e(\alpha)=f$ $\quad$ (Eq. 7 in the paper)

They skip the details of the minimization, but they do say they used Lagrange multipliers and give a closed form expression for the minimizer as $\alpha(\lambda_1, \lambda_2)$.

My question is, how can I reduce the constraints above to only two equations, so the minimizer will end up being a function of only two lambdas?

I thought about:

$a(\alpha) - 2b(\alpha) +c = 0$

and

$d(\alpha) - 2e(\alpha) +f = 0$

but I realize these don't necessarily imply the original constraints.

Answer 1

One way: \begin{align*} (a(\alpha) - c)^2 + (b(\alpha) - c)^2 = 0 &\text{ and} \\ (d(\alpha) - f)^2 + (e(\alpha) - f)^2 = 0 &\text{.} \end{align*}

This works if everything in sight is a real number or a real valued function. It does not work in the complex setting.