I want to solve some minimisation problem for a loss function of the form:
\begin{equation} \begin{split} x' &= \underset{x}{\arg\min} \;\; f(x)\\ & subject\;\;to\;\; c_i>c_j \;\; where\;\; i>j \end{split} \end{equation}
The issue is that there can be multiple such ranked constraints such as $c1<c2<c3<c4<c5...$ and to use a Lagrangian solution on this would involve taking all possible pairwaise combinations such as $\lambda_{k}(c_i-c_j)$. Is there a better way to optimise this? I am not sure how to search for this exactly on Google, so if someone could guide me some resources I could read it up myself as well.
As a side note for my specific problem it is most likely a convex optimisation problem (still working on it).