In
vector calculus, an invex function is a
differentiable function
from
to
for which there exists a vector valued function
such that
![{\displaystyle f(x)-f(u)\geq \eta (x,u)\cdot \nabla f(u),\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27048c660525cd401cdb80b9a9b200e604419c65)
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.
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.
[4] Consider a mathematical program of the form
where
and
are differentiable functions. Let
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
with respect to
if there exists a vector-valued function
defined on
such that
and
for all
.
[5] Note that, unlike invexity, Type I invexity is defined relative to a point
.
Theorem (Theorem 2.1 in
[4]): If
and
are Type I invex at a point
with respect to
, and the
Karush–Kuhn–Tucker conditions are satisfied at
, then
is a global minimizer of
over
.
E-invex function
Let
from
to
and
from
to
be an
-differentiable function on a nonempty open set
. Then
is said to be an E-invex function at
if there exists a vector valued function
such that
![{\displaystyle (f\circ E)(x)-(f\circ E)(u)\geq \nabla (f\circ E)(u)\cdot \eta (E(x),E(u)),\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f4a1ab4ab6ec2086cb33353adc0b6f7c6826446)
for all
and
in
.
E-invex functions were introduced by Abdulaleem as a generalization of differentiable
convex functions.
[6]
See also
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.