Invex function

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

: $f(x) - f(u) \geq \eta(x, u) \cdot \nabla f(u), \,$

for all x and u.

Invex functions were introduced by Hanson as a generalization of convex functions.{{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}} 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.{{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}}{{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}}

Hanson also showed that if the objective and the constraints of an optimization problem are invex with respect to the same function $\eta(x,u)$ , 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.{{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}} Consider a mathematical program of the form

$\begin{array}{rl}
\min & f(x)\\
\text{s.t.} & g(x)\leq0
\end{array}$

where $f:\mathbb{R}^n\to\mathbb{R}$ and $g:\mathbb{R}^n\to\mathbb{R}^m$ are differentiable functions. Let $F=\{x\in\mathbb{R}^n\;|\;g(x)\leq0\}$ 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 $\eta$ if there exists a vector-valued function $\eta$ defined on $F$ such that

$f(x)-f(x_0)\geq\eta(x)\cdot\nabla{f(x_0)}$

and

$-g(x_0)\geq\eta(x)\cdot\nabla{g(x_0)}$

for all $x\in{F}$ .{{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}} 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 $\eta$ , 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 $\mathbb{R}^n$ to $\mathbb{R}^{n}$ and $f$ from $\mathbb{M}$ to $\mathbb{R}$ be an $E$ -differentiable function on a nonempty open set $\mathbb{M} \subset \mathbb{R}^n$ . Then $f$ is said to be an E-invex function at $u$ if there exists a vector valued function $\eta$ such that

: $(f\circ E)(x) - (f\circ E)(u) \geq \nabla (f\circ E)(u) \cdot \eta(E(x), E(u)) , \,$

for all $x$ and $u$ in $\mathbb{M}$ .

E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions.{{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}}

E-type I Functions

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

$f_i(E(x)) - f_i(E(u)) \geq \nabla f_i(E(u)) \cdot \eta(E(x), E(u)),$

$-g_j(E(u)) \geq \nabla g_j(E(u)) \cdot \eta(E(x), E(u)).$

= Remark 1. =

If $f$ and $g$ are differentiable functions and $E(x) = x$ ( $E$ is an identity map), then the definition of E-type I functions{{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 }} reduces to the definition of type I functions introduced by Rueda and Hanson.{{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}}

Invex function

Type I invex functions

E-type I Functions

= Remark 1. =

See also

References

Further reading