  A central goal in designing clinical trials is to find the test that
maximizes power (or equivalently minimizes required sample size) for finding a
true research 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 policies 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 policies. This leads us to formulate the optimal policy
problem as an explicit optimization problem with objective and constraints
which describe its specific desiderata. We describe a complete solution for
deriving optimal policies for two hypotheses, which have desired monotonicity
properties, and are computationally simple. For some of the optimization
formulations this yields optimal policies that are identical to existing
policies, 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).
