# invex functions - optimality functions

For a general convex program, a feasible point is an optimal solution if and only if it lies in a hyperplane whose a normal vector is the gradient to the objective function at this point. Please suggest what will be the form of this result in case of an invex function involving a non-linear function eta.

For an convex function $f$, $x$ is an optimal solution if and only if $\langle \nabla f(x) , y-x \rangle \geq 0$ which explains the the fact that gradient of the objective function is the normal to the hyperplane at x.

For an invex function $f$, $x$ is an optimal solution if and only if $\langle \nabla f(x) , n(y,x) \rangle \geq 0$, where $n(y,x)$ is a nonlinear function.

Looks like in case of an invex function, the gradient of the objective at an optimal point must make an acute angle with all the non linear curves $n$ ?

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Isn't it the fact you're looking for? en.wikipedia.org/wiki/Invex_function Sentence about Ben-Israel and Mond. – Ilya Sep 26 '11 at 16:12
@Gortaur: I am looking for a geometrical interpretation of optimality conditions for differential invex functions, just like the ones provided for differentiable convex functions above. – Tav Sep 26 '11 at 16:27

The Wikipedia article says that if the objective and constraints are invex wrt the same $g(x,u)$, the Karush-Kuhn-Tucker conditions are sufficient for a global minimum. Geometrically the Karush-Kuhn-Tucker conditions say that the gradient of the objective is in the cone generated by the outward normals of the active constraints.