# Minimizing a functional constrained to maximizing another

Consider the following minimization problem $$$$\min_{\phi}\int_V d^3x\ f(\phi(x),\phi'(x)) \tag{1}$$$$ to be constrained by maximizing $$$$\max_{\phi}\int_V d^3x\ g(\phi(x)). \tag{2}$$$$ 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

$$$$\min_{\phi} \left\{\int_V d^3x\ f(\phi(x),\phi'(x)) - \int_V d^3x\ |g(\phi(x))|^2 \right\}, \tag{3}$$$$

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.

• en.wikipedia.org/wiki/Bilevel_optimization Jul 14, 2023 at 18:56
• Wow! This seems exactly what I am looking for! Still, I don't know how it works for my problem. Any similar examples? Jul 14, 2023 at 19:12