Invex function explained

f from R ^{n} to R for which there exists a vector valued function η such thatf(x) - f(u) \geq η (x, u) ⋅ \nabla f(u),

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

η (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.^[4] Consider a mathematical program of the form

\begin{array}{rl} min & f(x)\\ s.t. & g(x)\leq 0 \end{array}

where

f: R ^{n\toR} and g: R ^{n\toR} ^{m} are differentiable functions. Let F = \{x\in R ^{n | g(x)\leq0\}} 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 η if there exists a vector-valued function η defined on F such that f(x) - f(x _{0)\geqη(x) ⋅ \nabla{f(x} _{0)}}

and

- g(x _{0)\geqη(x) ⋅ \nabla{g(x} _{0)}}

for all

x\in{F} .^[5] 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 η , and the Karush–Kuhn–Tucker conditions are satisfied at x ^{*} , then x ^{*} is a global minimizer of f over F .

E-invex function

Let

E from R ^{n} to R ^{n} and f from M to R be an E -differentiable function on a nonempty open set M \subset R ^{n} . Then f is said to be an E-invex function at u if there exists a vector valued function η such that(f\circ E)(x) - (f\circ E)(u) \geq \nabla (f\circ E)(u) ⋅ η (E(x), E(u)) ,

for all

x and u in M .

E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions.^[6]

See also

• Convex function
• Pseudoconvex function
• Quasiconvex function

References

1. Hanson. Morgan A.. 1981. On sufficiency of the Kuhn-Tucker conditions. Journal of Mathematical Analysis and Applications. 80. 2. 545–550. 10.1016/0022-247X(81)90123-2. 0022-247X. 10338.dmlcz/141569. free.
2. Ben-Israel. A.. Mond. B.. 1986. What is invexity?. The ANZIAM Journal. en. 28. 1. 1–9. 10.1017/S0334270000005142. 1839-4078. free.
3. Craven. B. D.. Glover. B. M.. 1985. Invex functions and duality. Journal of the Australian Mathematical Society. en. 39. 1. 1–20. 10.1017/S1446788700022126. 0263-6115. free.
4. Hanson. Morgan A.. 1999. Invexity and the Kuhn–Tucker Theorem. Journal of Mathematical Analysis and Applications. 236. 2. 594–604. 10.1006/jmaa.1999.6484. 0022-247X. free.
5. Hanson. M. A.. Mond. B.. 1987. Necessary and sufficient conditions in constrained optimization. Mathematical Programming. en. 37. 1. 51–58. 10.1007/BF02591683. 206818360 . 1436-4646.
6. Abdulaleem. Najeeb. 2019. E-invexity and generalized E-invexity in E-differentiable multiobjective programming . ITM Web of Conferences. 24. 1. 01002. 10.1051/itmconf/20192401002 . free.

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.