Invex function

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


 * $$f(x) - f(u) \geq \eta(x, u) \cdot \nabla f(u), \, $$

for all x and u.

Invex functions were introduced by Hanson as a generalization of convex functions. 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

$$\begin{array}{rl} \min & f(x)\\ \text{s.t.} & g(x)\leq0 \end{array}$$

where $$f:\mathbb{R}^n\to\mathbb{R}$$ and $$g:\mathbb{R}^n\to\mathbb{R}^m$$are differentiable functions. Let $$F=\{x\in\mathbb{R}^n\;|\;g(x)\leq0\}$$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

$$f(x)-f(x_0)\geq\eta(x)\cdot\nabla{f(x_0)}$$

and

$$-g(x_0)\geq\eta(x)\cdot\nabla{g(x_0)}$$

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 $$\mathbb{R}^n$$ to $$\mathbb{R}^{n}$$ and $$f$$ from $$\mathbb{M}$$ to $$\mathbb{R}$$ be an $$E$$-differentiable function on a nonempty open set $$\mathbb{M} \subset \mathbb{R}^n$$. Then $$f$$ is said to be an E-invex function at $$u$$  if there exists a vector valued function $$\eta$$ such that


 * $$(f\circ E)(x) - (f\circ E)(u) \geq \nabla (f\circ E)(u) \cdot \eta(E(x), E(u)), \, $$

for all $$x$$ and $$u$$ in $$\mathbb{M}$$.

E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions.