All optimal points (for, say, optimizing a linear function) will lie on the extremal points of the feasible domain, many of which will be points where the constraint functions are not differentiable. In all cases you can turn nonlinear objective function optimization (say over f) into linear objective function optimization by adding a constraint f(x) ≤ t and moving t to the objective.

Now, I will agree that smooth optimization algorithms will work ok, but try optimizing abs(x) with GD; you'll find that the best possible error you can achieve (other than by sheer luck) will be ~O(L) where L is your stepsize.

Sorry I screwed up a bit when writing that: an optimal point exists which lies on the extremal points of the feasible domain. In many cases, it can be shown that the optimal point lies on the extremal points of the feasible domain (when the objective is linear)

This is not, in general, true for smooth functions so long as L is small enough (you can reach arbitrary accuracy with GD if L is smaller than ~ the reciprocal of the Lipchitz constant of a differentiable objective function but it need not be arbitrarily small).