  We prove the optimal $W^{2, \infty }$ regularity for fully nonlinear elliptic
equations with convex gradient constraints. We do not assume any regularity
about the constraints; so the constraints need not be $C^1$ or strictly convex.
We also show that the optimal regularity holds up to the boundary. Our approach
is to show that these elliptic equations with gradient constraints are related
to some fully nonlinear double obstacle problems. Then we prove the optimal
$W^{2, \infty }$ regularity for the double obstacle problems. In this process,
we also employ the monotonicity property for the second derivative of
obstacles, which we have obtained in a previous work.
