Minimization Interchange Theorem on Posets

TL;DR

This paper generalizes the minimization interchange theorem by replacing integration with monotone mappings between posets, extending classical results and providing new insights, including applications to the Choquet integral.

Contribution

It introduces a generalized interchange theorem for minimization and monotone mappings on posets, extending classical results and applying to the Choquet integral.

Findings

Recovered classical interchange results

Extended results to the Choquet integral

Provided theoretical insights into interchange mechanisms

Abstract

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.