Invex function

In vector calculus, an invex function is a differentiable function ${\displaystyle f}$ from ${\displaystyle \mathbb {R} ^{n}}$ to ${\displaystyle \mathbb {R} }$ for which there exists a vector valued function ${\displaystyle \eta }$ such that

${\displaystyle f(x)-f(u)\geq \eta (x,u)\cdot \nabla f(u),\,}$

for all x and u.

Invex functions were introduced by Hanson as a generalization of convex functions.^[1] Ben-Israel and Mond provided a simple proof that a function is invex if and only if every stationary point is a global minimum, a theorem first stated by Craven and Glover.^[2][3]

Hanson also showed that if the objective and the constraints of an optimization problem are invex with respect to the same function ${\displaystyle \eta (x,u)}$, then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.