I have a function $f(u(x), v(x))$ where $u(x)=g_1(x)$, $v(x) = g_2(x)$. so basically both function determined on same input $x$. I wonder is there a way to proof $f$ can reach (local) optima when $u(x) = a$, where $a$ is a constant value?
the difficulty here is $u(x)$ and $v(x)$ is not independent, suppose there's no way to extract exact $x$ from $u(x) = a$, is there exist a rigorous way to prove this hypothesis?