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. 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.

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. Consider a mathematical program of the form

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

where $f:\mathbb {R} ^{n}\to \mathbb {R}$ and $g:\mathbb {R} ^{n}\to \mathbb {R} ^{m}$ are differentiable functions. Let $F=\{x\in \mathbb {R} ^{n}\;|\;g(x)\leq 0\}$ 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 $x\in {F}$. 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.

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 reduces to the definition of type I functions introduced by Rueda and Hanson.

See also

• Convex function
• Pseudoconvex function
• Quasiconvex function

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.