SIAM Journal on Optimization, Volume 31, Issue 1, Page 425-447, January 2021.
We consider a generalization of the Bauer maximum principle. We work with tensorial products of convex measures set that are not 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 measure sets. Finally, the usual compactness assumption is replaced with the existence of an integral representation on the extreme points. We focus on the product of two different types of measure 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 is available 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.

中文翻译：

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乘积度量集上拟凸函数的优化

SIAM优化杂志，第31卷，第1期，第425-447页，2021年1月。
我们考虑了鲍尔极大值原理的推广。我们处理凸度量集的张量积，它们不一定紧凑，而是由其极端点生成。我们表明，在极值的有限混合的张量积上，在该乘积空间上的拟凸下半连续函数的最大值达到了。我们的工作是在三个不同方面对鲍尔最大原理的扩展。首先，我们仅假设目标函数是拟凸的。其次，优化是在作为度量集的乘积构建的空间上执行的。最后，通常的紧密度假设被极端点上的完整表示形式所代替。我们关注两种不同类型的量度集的乘积，即矩类和单峰矩类。这些类别的元素是满足广义矩约束的概率测度（分别为单峰概率测度）。我们证明了在极端点上的积分表示可用于此类空间，并且扩展到其张量积。从稳健的贝叶斯分析到计算机代码输出分位数的优化，我们给出了该定理的几种应用。