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

Optimization Of Quasi-convex Function Over Product Measure Sets