Optimal multiple testing under family-wise error control: elementary symmetric polynomials and a scalable algorithm

Simultaneously testing $K$ hypotheses while controlling the family-wise error rate is a fundamental problem in statistics. Existing procedures (Bonferroni, Holm, Hochberg, Hommel) provide valid control but sacrifice power, increasingly so as $K$ grows, because they base decisions on marginal $p$-value ranks rather than the joint likelihood. Rosset et al. (2022) formulated the most powerful family-wise-error-rate-controlling test as a dual program and proved the existence of an optimal dual vector $\mu^*$, but left its computation as an open problem. We solve this problem for $K$ exchangeable hypotheses. The key insight is that the family-wise error rate constraint coefficients $b_{l,k}(\vec{u})$ admit closed-form expressions through elementary symmetric polynomials of the likelihood-ratio values $g(u_1), \ldots, g(u_K)$. This algebraic structure implies a global monotonicity theorem: the target functions $F_\gamma(\mu) = {\rm FWER}_\gamma(\vec{D}^\mu)$ are simultaneously non-increasing in every component of $\mu$, for arbitrary $K$, which guarantees unique coordinate-wise roots and enables a bisection-based coordinate-descent algorithm with $O(\log \varepsilon^{-1})$ convergence rate. The relative power gain over Hommel's method grows from 15\% at $K{=}3$ to 84\% at $K{=}12$. Applications to replication studies, a clinical trial, and a replicability assessment illustrate both the power gains and the role of the exchangeability assumption.