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. 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 η ( x , u ) {\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. 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}} . Note that, unlike invexity, Type I invexity is defined relative to a point x 0 {\displaystyle x_{0}} . Theorem (Theorem 2.1 in): 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} .
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