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