Invex Function

In vector calculus, an invex function is a differentiable function $f$ from $\mathbb^n$ to $\mathbb$ 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

$\begin \min & f(x)\\ \text & g(x)\leq0 \end$

where $f:\mathbb^n\to\mathbb$ and $g:\mathbb^n\to\mathbb^m$are differentiable functions. Let $F=\$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

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

and

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

for all $x\in$. 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 $F$.

See also

* Convex function
* Pseudoconvex function
* Quasiconvex function

References

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.

{{Convex analysis and variational analysis

Convex analysis
Generalized convexity
Real analysis
Types of functions