Invex function

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

[math]\displaystyle{ f(x) - f(u) \geq \eta(x, u) \cdot \nabla f(u), \, }[/math]

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 [math]\displaystyle{ \eta(x,u) }[/math], 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

[math]\displaystyle{ \begin{array}{rl} \min & f(x)\\ \text{s.t.} & g(x)\leq0 \end{array} }[/math]

where [math]\displaystyle{ f:\mathbb{R}^n\to\mathbb{R} }[/math] and [math]\displaystyle{ g:\mathbb{R}^n\to\mathbb{R}^m }[/math]are differentiable functions. Let [math]\displaystyle{ F=\{x\in\mathbb{R}^n\;|\;g(x)\leq0\} }[/math]denote the feasible region of this program. The function [math]\displaystyle{ f }[/math] is a Type I objective function and the function [math]\displaystyle{ g }[/math] is a Type I constraint function at [math]\displaystyle{ x_0 }[/math]with respect to [math]\displaystyle{ \eta }[/math] if there exists a vector-valued function [math]\displaystyle{ \eta }[/math] defined on [math]\displaystyle{ F }[/math] such that

[math]\displaystyle{ f(x)-f(x_0)\geq\eta(x)\cdot\nabla{f(x_0)} }[/math]

and

[math]\displaystyle{ -g(x_0)\geq\eta(x)\cdot\nabla{g(x_0)} }[/math]

for all [math]\displaystyle{ x\in{F} }[/math]. Note that, unlike invexity, Type I invexity is defined relative to a point [math]\displaystyle{ x_0 }[/math].

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

E-invex function

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

[math]\displaystyle{ (f\circ E)(x) - (f\circ E)(u) \geq \nabla (f\circ E)(u) \cdot \eta(E(x), E(u)) , \, }[/math]

for all [math]\displaystyle{ x }[/math] and [math]\displaystyle{ u }[/math] in [math]\displaystyle{ \mathbb{M} }[/math].

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

See also

• Convex function
• Pseudoconvex function
• Quasiconvex function

Further reading

• 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.