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 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 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. 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 {egin{array}{rl}min &f(x)\{ ext{s.t.}}&g(x)leq 0end{array}}} where f : R n → R {displaystyle f:mathbb {R} ^{n} o mathbb {R} } and g : R n → R m {displaystyle g:mathbb {R} ^{n} o mathbb {R} ^{m}} are differentiable functions. Let F = { x ∈ R n | g ( x ) ≤ 0 } {displaystyle F={xin 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 abla {f(x_{0})}}