  A central goal in designing clinical trials is to find the test that
maximizes power (or equivalently minimizes required sample size) for finding a
false null hypothesis subject to the constraint of type I error. When there is
more than one test, such as in clinical trials with multiple endpoints, the
issues of optimal design and optimal procedures become more complex. In this
paper we address the question of how such optimal tests should be defined and
how they can be found. We review different notions of power and how they relate
to study goals, and also consider the requirements of type I error control and
the nature of the procedures. This leads us to an explicit optimization problem
with objective and constraints which describe its specific desiderata. We
present a complete solution for deriving optimal procedures for two hypotheses,
which have desired monotonicity properties, and are computationally simple. For
some of the optimization formulations this yields optimal procedures that are
identical to existing procedures, such as Hommel's procedure or the procedure
of Bittman et al. (2009), while for others it yields completely novel and more
powerful policies than existing ones. We demonstrate the nature of our novel
policies and their improved power extensively in simulation and on the APEX
study (Cohen et al., 2016).
