Some questions about Invexity

Question

Recently, I am looking into a non-convex optimization problem whose points satisfying KKT conditions can be obtained. Then the problem becomes how to decide whether the KKT conditions are sufficient for the global optimality of the solution. It is said on Wikipedia(link text) that for "invex function" the KKT is sufficient for optimality. By doing some searching, I was led to a concept called "invexity" and the book "Invexity and Optimization" (http://www.springerlink.com/content/978-3-540-78561-3#section=155972&page=1).

But, why this "invexity" research, which was first proposed in 1980's, is only confined in a small group of people. And the book is cited for only 4 times.

In my current understanding, invexity is a generalization of convexity, and has some very good properties as in convexity, which should be very attractive and should have drawn lots of people's attention.

Is it because this concept is not interesting, not useful? Or it is being studied under other names?

Answer 1

There are a lot of generalizations or variations of convexity, such as quasi-convexity, pseudo-convexity, semilocal convexity, semilocal quasi-convexity, semilocal pseudo-convexity, strict versions of these, strong versions of these, etc. There is a reason for the existence of each term, in that each makes the hypotheses of some theorem tighter or has some other benefit (such as invexity making the KKT conditions not just necessary but sufficient). I agree with you that invexity ought to be better known, but it may have gotten lost in all the other generalizations/variations out there.

Answer 2

I second Czenek's recommendation. Moreover, it could be helpful to know this cute little characterization of invexity (due to Craven and Grower, see also here):

Theorem: A differentiable function $f$ is invex if and only if every stationary point is a global minimum.

Proof: Let $f$ be invex and $u$ stationary. Then $\nabla f(u) = 0$ and hence $f(x)-f(u)\geq 0$ for all $x$. Conversely, let every stationary point be globally optimal. Then $$ \eta(x,u) = \begin{cases}0 & \text{if}\ \nabla f(u) = 0\\\\ \frac{f(x)-f(u)}{\|\nabla f(u)\|^2}\nabla f(u) & \text{else}\end{cases} $$ shows that $f$ is invex.

Hence, invexity is just another way to say the every stationary point is globally optimal. Since the latter property is fulfilled by quite "arbitrary" functions, one may conclude that invexity does not impose too many structure.

Answer 3

Worth to read: "A critical view on invexity" by Constantin Zalinescu to be found here.