  Multiple testing problems are a staple of modern statistical analysis. The
fundamental objective of multiple testing procedures is to reject as many false
null hypotheses as possible (that is, maximize some notion of power), subject
to controlling an overall measure of false discovery, like family-wise error
rate (FWER) or false discovery rate (FDR). In this paper we formulate multiple
testing of simple hypotheses as an infinite-dimensional optimization problem,
seeking the most powerful rejection policy which guarantees strong control of
the selected measure. In that sense, our approach is a generalization of the
optimal Neyman-Pearson test for a single hypothesis. We show that for
exchangeable hypotheses, for both FWER and FDR and relevant notions of power,
these problems can be formulated as infinite linear programs and can in
principle be solved for any number of hypotheses. We also characterize maximin
rules for complex alternatives, and demonstrate that such rules can be found in
practice, leading to improved practical procedures compared to existing
alternatives. We derive explicit optimal tests for FWER or FDR control for
three independent normal means. We find that the power gain over natural
competitors is substantial in all settings examined. Finally, we apply our
optimal maximin rule to subgroup analyses in systematic reviews from the
Cochrane library, leading to an increase in the number of findings while
guaranteeing strong FWER control against the one sided alternative.
