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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.

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

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