Adaptive enrichment designs involve preplanned rules for modifying enrollment criteria based on accruing data in a randomized trial. These designs can be useful when it is suspected that treatment effects may differ in certain subpopulations, such as those defined by a biomarker or risk factor at baseline. Two critical components of adaptive enrichment designs are the decision rule for modifying enrollment, and the multiple testing procedure. We provide a general method for simultaneously optimizing both of these components for two stage, adaptive enrichment designs. The optimality criteria are defined in terms of expected sample size and power, under the constraint that the familywise Type I error rate is strongly controlled. It is infeasible to directly solve this optimization problem since it is not convex. The key to our approach is a novel representation of a discretized version of this optimization problem as a sparse linear program. We apply advanced optimization tools to solve this problem to high accuracy, revealing new, optimal designs. This is joint work with Xingyuan (Ethan) Fang and Han Liu at Princeton University.