Invex function
Encyclopedia
In vector calculus, an invex function is a differentiable function ƒ from Rn to R for which there exists a vector valued function g 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.
Hanson also showed that if the objective and the constraints of an optimization problem are invex with respect to the same function g(x, u), then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.
A slight generalization of invex functions called Type 1 invex functions are the most general class of functions for which the Karush–Kuhn–Tucker conditions are necessary and sufficient for a global minimum .
for all x and u.
Invex functions were introduced by Hanson as a generalization of convex functions
Convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
. Ben-Israel and Mond provided a simple proof that a function is invex if and only if every stationary point
Stationary point
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
is a global minimum.
Hanson also showed that if the objective and the constraints of an optimization problem are invex with respect to the same function g(x, u), then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.
A slight generalization of invex functions called Type 1 invex functions are the most general class of functions for which the Karush–Kuhn–Tucker conditions are necessary and sufficient for a global minimum .