In vector calculus, an invex function is a differentiable function f from to \mathbb{R} for which there exists a vector valued function \eta such that for all x and u. Invex functions were introduced by Hanson as a generalization of convex functions. 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. Hanson also showed that if the objective and the constraints of an optimization problem are invex with respect to the same function \eta(x,u), then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.

Type I invex functions

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. Consider a mathematical program of the form where and are differentiable functions. Let denote the feasible region of this program. The function f is a Type I objective function and the function g is a Type I constraint function at x_0 with respect to \eta if there exists a vector-valued function \eta defined on F such that and for all x\in{F}. Note that, unlike invexity, Type I invexity is defined relative to a point x_0. Theorem (Theorem 2.1 in ): If f and g are Type I invex at a point x^* with respect to \eta, and the Karush–Kuhn–Tucker conditions are satisfied at x^, then x^ is a global minimizer of f over F.

E-invex function

Let E from to and f from \mathbb{M} to \mathbb{R} be an E-differentiable function on a nonempty open set. Then f is said to be an E-invex function at u if there exists a vector valued function \eta such that for all x and u in \mathbb{M}. E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions.

E-type I Functions

Let, and be an open E-invex set. A vector-valued pair (f, g), where f and g represent objective and constraint functions respectively, is said to be E-type I with respect to a vector-valued function, at u \in M, if the following inequalities hold for all :

Remark 1.

If f and g are differentiable functions and E(x) = x (E is an identity map), then the definition of E-type I functions reduces to the definition of type I functions introduced by Rueda and Hanson.