In vector calculus, an invex function is a differentiable function
for which there exists a vector valued function
such that 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
, 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
min
{\displaystyle {\begin{array}{rl}\min &f(x)\\{\text{s.t.
are differentiable functions.
denote the feasible region of this program.
The function
is a Type I objective function and the function
is a Type I constraint function at
if there exists a vector-valued function
[5] Note that, unlike invexity, Type I invexity is defined relative to a point
are Type I invex at a point
, and the Karush–Kuhn–Tucker conditions are satisfied at
is a global minimizer of
-differentiable function on a nonempty open set
is said to be an E-invex function at
if there exists a vector valued function
E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions.
be an open E-invex set.
A vector-valued pair
represent objective and constraint functions respectively, is said to be E-type I with respect to a vector-valued function
, if the following inequalities hold for all
are differentiable functions and
is an identity map), then the definition of E-type I functions[7] reduces to the definition of type I functions introduced by Rueda and Hanson.