In this chapter we prove Theorems 6.3.5 through 6.3.22 and their corollaries. Theorem 6.3.5 will be proved by a penalty function method, which we outline here. For simplicity, let f be a real valued differentiable function defined on an open set X in Rn. Consider the unconstrained problem of minimizing f on X . If f attains a minimum at a point x0 in X , then the necessary condition df(x0) = 0 holds, where df is the differential of f . This condition is obtained by making a perturbation x0 + εδx, where δx is arbitrary but fixed, and ε is sufficiently small so that x0+εδx is in X . Then since f is differentiable and attains a minimum at x0,

f(x0 + εδx)− f(x0) = df(x0)εδx+ θ(εδx) ≥ 0,

where θ(εδx)/(εδx) → 0 as ε → 0. In the rightmost inequality if we first divide through by ε > 0 and then let ε → 0, we get df(x0)δx ≥ 0. If we divide through by ε < 0 and then let ε→ 0, we get df(x0)δx ≤ 0. Since δx is arbitrary, we get df(x0) = 0.