Invex function

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

f ( x ) − f ( u ) ≥ η ( x , u ) ⋅ ∇ f ( u ) , {\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 η ( 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

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

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

f ( x ) − f ( x 0 ) ≥ η ( x ) ⋅ ∇ f ( x 0 ) {\displaystyle f(x)-f(x_{0})\geq \eta (x)\cdot \nabla {f(x_{0})}}

and

− g ( x 0 ) ≥ η ( x ) ⋅ ∇ g ( x 0 ) {\displaystyle -g(x_{0})\geq \eta (x)\cdot \nabla {g(x_{0})}}

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

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

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

( f ∘ E ) ( x ) − ( f ∘ E ) ( u ) ≥ ∇ ( f ∘ E ) ( u ) ⋅ η ( E ( x ) , E ( u ) ) , {\displaystyle (f\circ E)(x)-(f\circ E)(u)\geq \nabla (f\circ E)(u)\cdot \eta (E(x),E(u)),\,}

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

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

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

f i ( E ( x ) ) − f i ( E ( u ) ) ≥ ∇ f i ( E ( u ) ) ⋅ η ( 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)),}

− g j ( E ( u ) ) ≥ ∇ g j ( E ( u ) ) ⋅ η ( E ( x ) , E ( u ) ) . {\displaystyle -g_{j}(E(u))\geq \nabla g_{j}(E(u))\cdot \eta (E(x),E(u)).}

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