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.

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. 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)\leq0\} 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 . 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)). 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 ==References== ==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.

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