The essence of invexity

The notion of invexity was introduced into optimization theory by Hanson in 1981 as a very broad generalization of convexity. A smooth mathematical program of the form minimizef(x), subject tog(x) ≦ 0, isx ∈D ? ? ⁿ invex if there exists a function η:D ×D → ? ⁿ such that, for allx, u ∈D, $$\begin{gathered} f(x) - f(u) - f'(u)n(x,u) \geqq 0, \hfill \\ g(x) - g(u) - g'(u)n(x,u) \geqq 0. \hfill \\ \end{gathered}$$ The convex case corresponds of course to η(x, u)≡x?u; but, as Hanson showed, invexity is sufficient to imply both weak duality and that the Kuhn-Tucker conditions are sufficient for global optimality. It is shown here that elementary relaxations of the conditions defining invexity lead to modified invexity notions which are both necessary and sufficient for weak duality and Kuhn-Tucker sufficiency.