Basic Optimality Conditions Using the Normal Cone

Theorem 3.1 Consider the convex optimization problem (CP ). Then x¯ is a point of minimizer of (CP ) if and only if either of two conditions hold:

(i) f ′(x¯, d) ≥ 0, ∀ d ∈ TC(x¯) or, (ii) 0 ∈ ∂f(x¯) +NC(x¯).