Consider the following minimization problem \begin{equation} \min_{\phi}\int_V d^3x\ f(\phi(x),\phi'(x)) \tag{1} \end{equation} to be constrained by maximizing \begin{equation} \max_{\phi}\int_V d^3x\ g(\phi(x)). \tag{2} \end{equation} I wonder if I could be provided with a simple approach to the problem. If I wanted to use Lagrange multipliers, how would the original problem be? I don't know how to implement KKT conditions, since the constraint is not an equality or non-equality, but a maximization problem.
Also, I much prefer to combine the two into a single minimization problem, like the following simplified problem
\begin{equation} \min_{\phi} \left\{\int_V d^3x\ f(\phi(x),\phi'(x)) - \int_V d^3x\ |g(\phi(x))|^2 \right\}, \tag{3} \end{equation}
which since we know that the second term is positive definite, does the work quite as I desire. (By the way, is it a conventional Lagrange multiplier problem with $\lambda=1$?) Since I want to keep the generality of the problem, I make no assumptions on $g$ (or, the integrand). But if any are needed, please let me know.