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 ${\displaystyle F}$.

E-invex function

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

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

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

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

E-type I Functions

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

${\displaystyle f_{i}(E(x))-f_{i}(E(u))\geq \nabla f_{i}(E(u))\cdot \eta (E(x),E(u)),}$

${\displaystyle -g_{j}(E(u))\geq \nabla g_{j}(E(u))\cdot \eta (E(x),E(u)).}$

Remark 1.

If ${\displaystyle f}$ and ${\displaystyle g}$ are differentiable functions and ${\displaystyle E(x)=x}$ (${\displaystyle E}$ is an identity map), then the definition of E-type I functions^[7] reduces to the definition of type I functions introduced by Rueda and Hanson.^[8]

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. S2CID 206818360.
6. Abdulaleem, Najeeb (2019). "E-invexity and generalized E-invexity in E-differentiable multiobjective programming". ITM Web of Conferences. 24 (1) 01002. doi:10.1051/itmconf/20192401002.
7. Abdulaleem, Najeeb (2023). "Optimality and duality for $ E $-differentiable multiobjective programming problems involving $ E $-type Ⅰ functions". Journal of Industrial and Management Optimization. 19 (2): 1513. doi:10.3934/jimo.2022004. ISSN 1547-5816.
8. Rueda, Norma G; Hanson, Morgan A (1988-03-01). "Optimality criteria in mathematical programming involving generalized invexity". Journal of Mathematical Analysis and Applications. 130 (2): 375–385. doi:10.1016/0022-247X(88)90313-7. ISSN 0022-247X.

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.