What does it mean to have a lone plus sign in the exponent/superscript (Modified Weiszfeld algorithm)

Question

I was skimming through Vardi & Zhang's paper on "A modified Weiszfeld algorithm for the Fermat-Weber location problem".

It is available at http://www.stat.rutgers.edu/home/cunhui/papers/43.pdf

If you open the paper and scroll down to page 3, you'll find equation 12, which I found to be odd. There is a lone plus sign in the exponent/superscript. Does anyone know what that means? Could it be that this simply a typo and that the "+" sign refers to an addition operation instead?

Many thanks for your help! Nicolas

Answer 1

A similar notation occurs in Equation (19), which suggests it's not a typo. In both instances, there is a convention mentioned in the following text, modifying the computation under certain circumstances. I suspect the + notation is just an indication that the expression is not always to be evaluated literally.

Answer 2

I have come across this notation in a different context (page 80, eq. (4.40)), where it was defined as

$$ x^+ \equiv \max(x,0) $$

And correspondingly

$$ x^- \equiv \min(x,0) $$

That is, the $+$ or $-$ indicates that the value is clamped to non-negative or non-positive numbers. Next to the other term in that equation, it seems likely that the same meaning is intended here.