Invex function

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

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

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 ${\displaystyle \eta (x,u)}${\displaystyle \eta (x,u)}, then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.

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

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

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

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

and

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

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

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

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

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

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

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

Let ${\displaystyle E:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}${\displaystyle E:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}, and ${\displaystyle M\subset \mathbb {R} ^{n}}${\displaystyle M\subset \mathbb {R} ^{n}}be an open E-invex set. A vector-valued pair ${\displaystyle (f,g)}${\displaystyle (f,g)}, where ${\displaystyle f}${\displaystyle f} and ${\displaystyle g}${\displaystyle g} represent objective and constraint functions respectively, is said to be E-type I with respect to a vector-valued function ${\displaystyle \eta :M\times M\to \mathbb {R} ^{n}}${\displaystyle \eta :M\times M\to \mathbb {R} ^{n}}, at ${\displaystyle u\in M}${\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 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 f_{i}(E(x))-f_{i}(E(u))\geq \nabla f_{i}(E(u))\cdot \eta (E(x),E(u)),}

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

If ${\displaystyle f}${\displaystyle f} and ${\displaystyle g}${\displaystyle g} are differentiable functions and ${\displaystyle E(x)=x}${\displaystyle E(x)=x} (${\displaystyle E}${\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]

• Convex function
• Pseudoconvex function
• Quasiconvex function

• 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
• Generalized convexity
• Real analysis
• Types of functions