In vector calculus, an invex function is a differentiable function 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, a theorem first stated by Craven and Glover.

Hanson also showed that if the objective and the constraints of an optimization problem, 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

is a Type I objective function is a Type I constraint function such that

and

..

Theorem (Theorem 2.1 in):, and the Karush–Kuhn–Tucker conditions.

E-invex function

such that

.

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

E-type I Functions

represent objective and constraint functions respectively, is said to be E-type I:

Remark 1.

is an identity map), then the definition of E-type I functions reduces to the definition of type I functions introduced by Rueda and Hanson.

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.