We give global optimality conditions expressed in terms of a function $\phi $ which satisfies conditions related to the Hamilton–Jacobi equation. Thus our results are in the spirit of sufficient conditions for optimality associated with Caratheodory in the calculus of variations, and of the verification theorems of optimal control theory. The novelty here is that $\phi $ is permitted to be merely Lipschitz continuous. A weakest hypothesis, strong calmness, is provided under which our results apply. Evidence that the strong calmness hypothesis is a reasonable one is presented elsewhere in the literature.