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 , then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.
A slight generalization of invex functions called Type I invex functions are the most general class of functions for which the Karush–Kuhn–Tucker conditions are necessary and sufficient for a global minimum.[4] Consider a mathematical program of the form
where and are differentiable functions. Let denote the feasible region of this program. The function is a Type Iobjective function and the function is a Type I constraint functionatwith respect to if there exists a vector-valued function defined on such that
and
for all .[5] Note that, unlike invexity, Type I invexity is defined relative to a point .
Theorem (Theorem 2.1 in[4]):If and are Type I invex at a point with respect to , and the Karush–Kuhn–Tucker conditions are satisfied at , then is a global minimizer of over .
Let from to and from to be an -differentiable function on a nonempty open set . Then is said to be an E-invex function at if there exists a vector valued function such that
for all and in.
E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions.[6]