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In optimization, often times people like to define a so-called extended (real) valued function, that is, a function that takes on the value $\infty$ outside of its domain.

But I have noticed that people tend to downplay the distinction between an extended valued function and its unextended counterpart. For example, in Boyd's text, on page 68, it reads:

In this book we will use the same symbol to denote a convex function and its extension, whenever there is no harm from the ambiguity. convex functions are implicitly extended, i.e., are defined as $\infty$ outside their domains.

However, in the rest of the text, extended value function rarely comes up and functions are almost always denoted as $f: \mathbb{R}^n \to \mathbb{R}$ instead of $f: \mathbb{R}^n \to \mathbb{R}\cup\{\infty\}$. (why not??) I wonder if it would be better to always stick with the un-extended version instead to avoid this ambiguity.

Here is another argument as to why I think it would be better just not to talk about the notion of the extended value function. These are some pros and cons of using extended valued functions from my understanding:

Pros:

  • Do not have to specify domain of variables in definition of convexity

  • Can represent certain functions such as the indicator function $I_{\mathcal{C}}$

Cons:

  • Infinite arithmetic when defining convex functions

  • The extended real line lacks good properties as compared to $\mathbb{R}$ in a topological sense

It seems to infinite arithmetics is a heavy price to pay for using the extended valued functions. We are extending the arithmetic system, albeit a trivial and intuitively acceptable extension...

So are there strong arguments as to why one should or should not use extended value functions? How and when should one use this concept?

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  • $\begingroup$ "The extended real line lacks good properties as compared to ℝ in a topological sense." How's that? Personally, I've always preferred compact spaces. :-) $\endgroup$ – saulspatz Jan 27 '18 at 4:10
  • $\begingroup$ @saulspatz There was a post discussing this a few years back, I don't remember the details, just remembered not to use the extended real line in public, you know, don't want to be "that" guy. math.stackexchange.com/questions/687220/… $\endgroup$ – Shamisen Expert Jan 27 '18 at 4:14
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One of the fundamental transformations in Convex Analysis is the Fenchel transform of a given $f$, defined by $$f^*(u) := \sup_{x\in X}\big(\langle u,x\rangle -f(x)\big).$$ This function is defined on the entire space $X$ (or $X^*$ if you work outside Hilbert space) and $+\infty$ is a natural possible value. Just compute it for a linear function - you get the indicator of a singleton. If you want to lug the domain around, as was done in early papers (Moreau etc.), then you get an ultra klunky theory...

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  • $\begingroup$ Hi can you provide a reference $\endgroup$ – Shamisen Expert Jan 27 '18 at 4:11
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    $\begingroup$ An early one is Rockafellar's Convex Analysis and a more recent one is Bauschke-Combettes' Convex Analysis and Monotone Operator Theory in Hilbert Spaces. $\endgroup$ – max_zorn Jan 27 '18 at 6:05
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Many properties regarding convexity of functions or minimization problems do not require convexity of $f$ on entire space but on a subset, like in convex optimization, function need to be convex on feasible set. So in order to avoid everytime mentioning "THIS PROPERTY HOLDS RELATIVE TO SUBSET C" authors prefer to initially work with extended real valued function. Then you can assume $f$ is convex on $C := domf$ and it take infinity outside of $C$, so $f$ is convex every where

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