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In [[vector calculus]], an '''invex function''' is a [[differentiable function]] <math>f</math> from <math>\mathbb{R}^n</math> to <math>\mathbb{R}</math> for which there exists a vector valued function <math>\eta</math> such that

:<math>f(x) - f(u) \geq \eta(x, u) \cdot \nabla f(u), \, </math>

for all ''x'' and ''u''.

Invex functions were introduced by Hanson as a generalization of [[convex function]]s.<ref>{{Cite journal|last=Hanson|first=Morgan A.|date=1981|title=On sufficiency of the Kuhn-Tucker conditions|journal=Journal of Mathematical Analysis and Applications|volume=80|issue=2|pages=545–550|doi=10.1016/0022-247X(81)90123-2|issn=0022-247X|hdl=10338.dmlcz/141569|hdl-access=free}}</ref> 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.<ref>{{Cite journal|last1=Ben-Israel|first1=A.|last2=Mond|first2=B.|date=1986|title=What is invexity?|journal=The ANZIAM Journal|language=en|volume=28|issue=1|pages=1–9|doi=10.1017/S0334270000005142|issn=1839-4078|doi-access=free}}</ref><ref>{{Cite journal|last1=Craven|first1=B. D.|last2=Glover|first2=B. M.|date=1985|title=Invex functions and duality|journal=Journal of the Australian Mathematical Society|language=en|volume=39|issue=1|pages=1–20|doi=10.1017/S1446788700022126|issn=0263-6115|doi-access=free}}</ref>

Hanson also showed that if the objective and the constraints of an [[optimization problem]] are invex with respect to the same function <math>\eta(x,u) </math>, 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.<ref name=":0">{{Cite journal|last=Hanson|first=Morgan A.|date=1999|title=Invexity and the Kuhn–Tucker Theorem|journal=Journal of Mathematical Analysis and Applications|volume=236|issue=2|pages=594–604|doi=10.1006/jmaa.1999.6484|issn=0022-247X|doi-access=free}}</ref> Consider a mathematical program of the form

<math>\begin{array}{rl}
\min & f(x)\\
\text{s.t.} & g(x)\leq0
\end{array}</math>

where <math>f:\mathbb{R}^n\to\mathbb{R}</math> and <math>g:\mathbb{R}^n\to\mathbb{R}^m</math> are differentiable functions. Let <math>F=\{x\in\mathbb{R}^n\;|\;g(x)\leq0\}</math> denote the feasible region of this program. The function <math>f</math> is a '''Type I''' '''objective function''' and the function <math>g</math> is a '''Type I constraint function''' at <math>x_0</math> with respect to <math>\eta</math> if there exists a vector-valued function <math>\eta</math> defined on <math>F</math> such that

<math>f(x)-f(x_0)\geq\eta(x)\cdot\nabla{f(x_0)}</math>

and

<math>-g(x_0)\geq\eta(x)\cdot\nabla{g(x_0)}</math>

for all <math>x\in{F}</math>.<ref>{{Cite journal|last1=Hanson|first1=M. A.|last2=Mond|first2=B.|date=1987|title=Necessary and sufficient conditions in constrained optimization|journal=Mathematical Programming|language=en|volume=37|issue=1|pages=51–58|doi=10.1007/BF02591683|s2cid=206818360 |issn=1436-4646}}</ref> Note that, unlike invexity, Type I invexity is defined relative to a point <math>x_0</math>.

'''Theorem (Theorem 2.1 in'''<ref name=":0" />'''):''' If <math>f </math> and <math>g </math> are Type I invex at a point <math>x^* </math> with respect to <math>\eta </math>, and the [[Karush–Kuhn–Tucker conditions]] are satisfied at <math>x^* </math>, then <math>x^* </math> is a global minimizer of <math>f </math> over <math>F </math>.

== E-invex function ==
Let <math>E</math> from <math>\mathbb{R}^n</math> to <math>\mathbb{R}^{n}</math> and <math>f</math> from <math>\mathbb{M}</math> to <math>\mathbb{R}</math> be an <math>E</math>-differentiable function on a nonempty open set <math>\mathbb{M} \subset \mathbb{R}^n</math>. Then <math>f</math> is said to be an E-invex function at <math>u</math> if there exists a vector valued function <math>\eta</math> such that

:<math>(f\circ E)(x) - (f\circ E)(u) \geq \nabla (f\circ E)(u) \cdot \eta(E(x), E(u)) , \, </math>

for all <math>x</math> and <math>u</math> in <math>\mathbb{M}</math>.

E-invex functions were introduced by Abdulaleem as a generalization of differentiable [[convex function]]s.<ref>{{Cite journal|last=Abdulaleem|first=Najeeb|date=2019|title=''E''-invexity and generalized ''E''-invexity in ''E''-differentiable multiobjective programming |journal=ITM Web of Conferences|volume=24|issue=1|article-number=01002|doi=10.1051/itmconf/20192401002 |doi-access=free}}</ref>

== E-type I Functions ==
Let <math>E: \mathbb{R}^n \to \mathbb{R}^n </math>, and <math>M \subset \mathbb{R}^n </math>be an open E-invex set. A vector-valued pair <math>(f, g) </math>, where <math>f</math> and <math>g</math> represent objective and constraint functions respectively, is said to be '''E-type I''' with respect to a vector-valued function <math>\eta: M \times M \to \mathbb{R}^n </math>, at <math>u \in M </math>, if the following inequalities hold for all <math>x \in F_E=\{x\in\mathbb{R}^n\;|\;g(E(x))\leq 0\} </math>:

<math>f_i(E(x)) - f_i(E(u)) \geq \nabla f_i(E(u)) \cdot \eta(E(x), E(u)), </math>

<math>-g_j(E(u)) \geq \nabla g_j(E(u)) \cdot \eta(E(x), E(u)). </math>

=== Remark 1. ===
If <math>f</math> and <math>g</math> are differentiable functions and <math>E(x) = x</math> (<math>E</math> is an identity map), then the definition of E-type I functions<ref>{{Cite journal |last=Abdulaleem |first=Najeeb |date=2023 |title=Optimality and duality for $ E $-differentiable multiobjective programming problems involving $ E $-type Ⅰ functions |url=https://www.aimsciences.org/article/doi/10.3934/jimo.2022004 |journal=Journal of Industrial and Management Optimization |volume=19 |issue=2 |pages=1513 |doi=10.3934/jimo.2022004 |issn=1547-5816|doi-access=free }}</ref> reduces to the definition of type I functions introduced by Rueda and Hanson.<ref>{{Cite journal |last=Rueda |first=Norma G |last2=Hanson |first2=Morgan A |date=1988-03-01 |title=Optimality criteria in mathematical programming involving generalized invexity |url=https://linkinghub.elsevier.com/retrieve/pii/0022247X88903137 |journal=Journal of Mathematical Analysis and Applications |volume=130 |issue=2 |pages=375–385 |doi=10.1016/0022-247X(88)90313-7 |issn=0022-247X}}</ref>

==See also==
* [[Convex function]]
* [[Pseudoconvex function]]
* [[Quasiconvex function]]

==References==

<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.

{{Convex analysis and variational analysis}}

[[Category:Convex analysis]]
[[Category:Generalized convexity]]
[[Category:Real analysis]]
[[Category:Types of functions]]

Invex function - Revision history

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