# 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 $x, u$, $f(x) - f(u) \geq \eta(x, u) \nabla f(u)$ .

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?

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You're right that if $f$ is invex then any stationary point is a global minimum. Other classes of functions, such as convex and quasiconvex functions, also share this property, though.
However, a key aspect of invex functions is that the relationship goes the other direction, too; i.e., if $f$ is differentiable and all stationary points of $f$ are global minima then $f$ must be invex. (See here for a proof.) So invex functions are exactly that class of functions that behave like our calculus students tend to think functions ought to behave when minimizing: Just find where $\nabla f = {\bf 0}$, and you're done. No need to test for local minima or saddle points or maxima or anything else. :)