We are concerned with the optimal design of composite materials with periodic microstructures. A homogenization approach is applied to obtain a computationally feasible macromodel. The microstructural geometrical details of the microcells are considered as design variables. The goal is to find the best material-and-shape combination in order to achieve optimal performance of the material with respect to a mechanical merit function. The resulting PDE constrained optimization problem is based on the equations of elasticity as state equations and additional technically motivated equality and inequality constraints. The numerical solution uses an all-at-once approach featuring an adaptive path-following prima-dual interior point method. Numerical results illustrate the performance of the algorithm.