Suppose that x* is a local solution of maxxG f(x). Then H+(x) D(x) = . Unfortunately,

Question:

Suppose that x* is a local solution of maxx∈G f(x). Then H+(x*) ∩ D(x*) = ϕ.
Unfortunately, the set of feasible directions does not exhaust the set of relevant perturbations, and we need to consider a broader class. Suppose that x ∈ D(x*) with x* + ax ∈ G for every 0 < a < . Let ak 1/k. Then the sequence xk = x* + akx ∈ G converges to x*. Furthermore xk - x* = akx and therefore the sequence xk - x*) = ak converges trivially to x. We say that xk converges to x* from the direction x (figure 5.9).