Minimization Interchange Theorem on Posets

Interchange theorems between minimization and integration are useful in optimization, especially in optimal control and in stochastic optimization. In this article, we establish a generalized minimization interchange theorem, where integration is replaced by a monotone mapping between posets (partially ordered sets). As an application, we recover, and slightly extend, classical results from the literature, and we tackle the case of the Choquet integral. Our result provides insight on the mechanisms behind existing interchange results.