I have a simple question. We know that functions where every stationary point is a global minimum are invex functions. Is there a name for functions where every local minimum is a global minimum?

And also can we naturally put convex functions into this scheme? As I see, claiming that it has only one global minimum that is the only stationary point is not enough.

Thank you!

Also, there is no tag for invexity

  • $\begingroup$ you are thinking of a function which is not convex and yet every local minimum is a global minimum? $\endgroup$ – Carlo Beenakker Apr 4 '17 at 17:23
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    $\begingroup$ There days people are calling such functions: "functions without bad local minima", though I must admit, that is not much of name :-) $\endgroup$ – Suvrit Apr 4 '17 at 17:35
  • $\begingroup$ @Carlo yes, it can be very crazy $\endgroup$ – Eugene Apr 4 '17 at 19:38
  • $\begingroup$ @Suvrit I like the phrase actually $\endgroup$ – Eugene Apr 4 '17 at 19:38
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    $\begingroup$ When local≠global minimum, the local minimum has been called as "spurious local minimum". See for instance the arXiv pre-prints 1605.07272 and 1704.00708. $\endgroup$ – Shamisen Apr 6 '17 at 2:14

I don't know of a specific term that describes this property, but it is different from convexity and invexity:

This function has a single global mimumum, but it has multiple stationary points, and therefore it is not invex:

$$ f(x) = 3 x^4 - 4 x^3 $$

And also can we naturally put convex functions into this scheme?

For a convex function, a local minumum will also be global.

However, the inverse is not true. A function may have only one local minimum, but be non-convex, for example

$$ f(x) = - e^{-x^2} $$

is not convex, but has only one local minumum (and also a single stationary point)


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