Invex function

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

$f(x) - f(u) \geqq g(x, u) \cdot \nabla f(u), \,$

for all x and u.

Invex functions were introduced by Hanson ^[1] as a generalization of convex functions. Ben-Israel and Mond ^[2] 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.^[3]

See also

• Convex function
• Pseudoconvex function
• Quasiconvex function

References

1. M.A. Hanson, On sufficiency of the Kuhn–Tucker conditions, J. Math. Anal. Appl. 80, pp. 545–550 (1981)
2. Ben-Israel, A. and Mond, B., What is invexity?, The ANZIAM Journal 28, pp. 1–9 (1986)
3. M.A. Hanson, Invexity and the Kuhn-Tucker Theorem, J. Math. Anal. Appl. vol. 236, pp. 594–604 (1999)

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.