Optimization Of Quasi-convex Function Over Product Measure Sets

TL;DR

This paper extends the Bauer maximum principle to optimize quasi-convex functions over tensor products of measure sets, allowing for non-compact spaces with integral representations, with applications in Bayesian analysis and computer code quantile optimization.

Contribution

It generalizes the Bauer maximum principle to non-compact measure spaces formed as tensor products, using integral representations on extreme points for quasi-convex functions.

Findings

Maximum of quasi-convex functions is attained on tensor products of finite mixtures of extreme points.

The integral representation extends to tensor products of measure sets like moment and unimodal moment classes.

Applications include robust Bayesian analysis and computer code output quantile optimization.

Abstract

We consider a generalization of the Bauer maximum principle. We work with tensorial products of convex measures sets, that are non necessarily compact but generated by their extreme points. We show that the maximum of a quasi-convex lower semicontinuous function on this product space is reached on the tensorial product of finite mixtures of extreme points. Our work is an extension of the Bauer maximum principle in three different aspects. First, we only assume that the objective functional is quasi-convex. Secondly, the optimization is performed over a space built as a product of measures sets. Finally, the usual compactness assumption is replaced with the existence of an integral representation on the extreme points. We focus on product of two different types of measures sets, called the moment class and the unimodal moment class. The elements of these classes are probability measures (respectively unimodal probability measures) satisfying generalized moment constraints. We show that an integral representation on the extreme points stands for such spaces and that it extends to their tensorial product. We give several applications of the Theorem, going from robust Bayesian analysis to the optimization of a quantile of a computer code output.