On the expected optimal value and the optimal expected value

Approaches to stochastic optimization have followed a variety of modeling philosophies, but little has been done to systematically compare different models found in the literature. This article is concerned with the basic concepts (and a comparison between them) underlying optimality under uncertainty, which is ubiquitous in all realistic problems of science and engineering. Specifically, it discusses two basic ideas—the minimum (maximum) expected value criterion and the expected minimum (maximum) value criterion—in a theoretical context. Illustrative applications are presented to justify the theoretical results.