  In confirmatory clinical trials with small sample sizes, hypothesis tests
based on asymptotic distributions are often not valid and exact non-parametric
procedures are applied instead. However, the latter are based on discrete test
statistics and can become very conservative, even more so, if adjustments for
multiple testing as the Bonferroni correction are applied. We propose improved
exact multiple testing procedures for the setting where two parallel groups are
compared in multiple binary endpoints. Based on the joint conditional
distribution of test statistics of Fisher's exact tests, optimal rejection
regions for intersection hypotheses tests are constructed. To efficiently
search the large space of possible rejection regions, we propose an
optimization algorithm based on constrained optimization and integer linear
programming. Depending on the optimization objective, the optimal test yields
maximal power under a specific alternative, maximal exhaustion of the nominal
type I error rate, or the largest possible rejection region controlling the
type I error rate. Applying the closed testing principle, we construct
optimized multiple testing procedures with strong familywise error rate
control. Furthermore, we propose a greedy algorithm for nearly optimal tests,
which is computationally more efficient. We numerically compare the
unconditional power of the optimized procedure with alternative approaches and
illustrate the optimal tests with a clinical trial example in a rare disease.
