invex functions and their usefulness? An invex function $f$ is a differentiable function from $\Bbb R^n$ to $\Bbb R$ that for some function $\eta : \Bbb R^n \times \Bbb R^n \to \Bbb R^n$ satisfies for all $x, u$, $f(x) - f(u) \geq \eta(x, u) \cdot \nabla f(u)$ .
The Wikipedia article on invex functions can be found here.
Since there are no restrictions on $\eta$, give a pair of vectors $x$ and $u$, I can always find a vector $z$ such that the dot product of this vector with the gradient vector at $u$ takes any real value (assuming the gradient vector is not degenerate or unbounded).  Thus, the only bite this definition has is at points where the gradient is $0$.  In fact, any stationary point i.e. point with gradient of $0$ must be a global minimum.  So invex functions are just those functions that have all stationary points as global minima.
Have I misunderstood something?

There has in fact been some controversy over the usefulness of invex functions at all.
This article is interesting A critical view on invexity
As far as I can tell, the criticism is justified and the definition is vacuous. Please can someone prove me wrong by providing a concrete example of a novel, interesting theorem which cannot be easily proven without using the concept of invexity!
