Invex function

In vector calculus, an invex function is a differentiable function ${\displaystyle f}$ from ${\displaystyle \mathbb {R} ^{n}}$ to ${\displaystyle \mathbb {R} }$ for which there exists a vector valued function ${\displaystyle \eta }$ such that

${\displaystyle f(x)-f(u)\geq \eta (x,u)\cdot \nabla f(u),\,}$

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 ${\displaystyle \eta (x,u)}$, 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

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

where ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ and ${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$are differentiable functions. Let ${\displaystyle F=\{x\in \mathbb {R} ^{n}\;|\;g(x)\leq 0\}}$denote the feasible region of this program. The function ${\displaystyle f}$ is a Type I objective function and the function ${\displaystyle g}$ is a Type I constraint function at ${\displaystyle x_{0}}$with respect to ${\displaystyle \eta }$ if there exists a vector-valued function ${\displaystyle \eta }$ defined on ${\displaystyle F}$ such that

${\displaystyle f(x)-f(x_{0})\geq \eta (x)\cdot \nabla {f(x_{0})}}$

and

${\displaystyle -g(x_{0})\geq \eta (x)\cdot \nabla {g(x_{0})}}$

for all ${\displaystyle x\in {F}}$.[5] Note that, unlike invexity, Type I invexity is defined relative to a point ${\displaystyle x_{0}}$.

Theorem (Theorem 2.1 in[4]): If ${\displaystyle f}$ and ${\displaystyle g}$ are Type I invex at a point ${\displaystyle x^{*}}$with respect to ${\displaystyle \eta }$, and the Karush–Kuhn–Tucker conditions are satisfied at ${\displaystyle x^{*}}$, then ${\displaystyle x^{*}}$is a global minimizer of ${\displaystyle f}$ over ${\displaystyle F}$.

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. doi:10.1016/0022-247X(81)90123-2. hdl:10338.dmlcz/141569. ISSN 0022-247X.
2. Ben-Israel, A.; Mond, B. (1986). "What is invexity?". The ANZIAM Journal. 28 (1): 1–9. doi:10.1017/S0334270000005142. ISSN 1839-4078.
3. Craven, B. D.; Glover, B. M. (1985). "Invex functions and duality". Journal of the Australian Mathematical Society. 39 (1): 1–20. doi:10.1017/S1446788700022126. ISSN 0263-6115.
4. Hanson, Morgan A. (1999). "Invexity and the Kuhn–Tucker Theorem". Journal of Mathematical Analysis and Applications. 236 (2): 594–604. doi:10.1006/jmaa.1999.6484. ISSN 0022-247X.
5. Hanson, M. A.; Mond, B. (1987). "Necessary and sufficient conditions in constrained optimization". Mathematical Programming. 37 (1): 51–58. doi:10.1007/BF02591683. ISSN 1436-4646.

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.