Exploiting Known Structures to Approximate Normal Cones

The normal cone to a constraint set plays a key role in optimization theory, algorithms, and applications. We consider the question of how to approximate the normal cone to a set under the assumption that the set is provided through an oracle function or collection of oracle functions, but contains some exploitable structure. We provide a new simplex gradient based approximation technique that works for sets of the form S = {x ∣∣gi(x) ≤ 0, i = 1, . . . , N}, where each gi : IR n → IR is unknown and provided by an oracle. We further present novel results showing that, under a non-degeneracy condition, approximating normal cones to intersections of sets is possible by taking sums of approximations. Finally, we provide numerical results that exemplify the accuracy of the simplex gradient approximation when it is applicable, and the fail of this technique then a linear independence constraint qualification is not met.