We consider the Euler equations of incompressible fluids and attempt to solve the initial value problem with the help of a concave maximization problem. We show that this problem, which shares a similar structure with the optimal transport problem with quadratic cost, in its "Benamou-Brenier" formulation, always admits a relaxed solution that can be interpreted in terms of $sub-solution$ of the Euler equations in the sense of convex integration theory. Moreover, any smooth solution of the Euler equations can be recovered from this maximization problem, at least for short times.