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

${\displaystyle {\begin{array}{rl}\min &f(x)\\{\text{s.t.}}&g(x)\leq 0\end{array}}}$ 

where ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$  and ${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ are differentiable functions. Let ${\displaystyle F=\{x\in \mathbb {R} ^{n}\;|\;g(x)\leq 0\}}$ denote the feasible region of this program. The function ${\displaystyle f}$  is a Type I objective function and the function ${\displaystyle g}$  is a Type I constraint function at ${\displaystyle x_{0}}$ with respect to ${\displaystyle \eta }$  if there exists a vector-valued function ${\displaystyle \eta }$  defined on ${\displaystyle F}$  such that

${\displaystyle f(x)-f(x_{0})\geq \eta (x)\cdot \nabla {f(x_{0})}}$ 

and

${\displaystyle -g(x_{0})\geq \eta (x)\cdot \nabla {g(x_{0})}}$ 

for all ${\displaystyle x\in {F}}$ .^[5] Note that, unlike invexity, Type I invexity is defined relative to a point ${\displaystyle x_{0}}$ .

Theorem (Theorem 2.1 in^[4]): If ${\displaystyle f}$  and ${\displaystyle g}$  are Type I invex at a point ${\displaystyle x^{*}}$ with respect to ${\displaystyle \eta }$ , and the Karush–Kuhn–Tucker conditions are satisfied at ${\displaystyle x^{*}}$ , then ${\displaystyle x^{*}}$ is a global minimizer of ${\displaystyle f}$  over ${\displaystyle F}$ .

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

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

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

E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions.^[6]

• Convex function
• Pseudoconvex function
• Quasiconvex function

• S. K. Mishra and G. Giorgi, Invexity and optimization, Nonconvex Optimization and Its Applications, Vol. 88, Springer-Verlag, Berlin, 2008.
• S. K. Mishra, S.-Y. Wang and K. K. Lai, Generalized Convexity and Vector Optimization, Springer, New York, 2009.