I have stumbled across this blob of text when reading my textbook, and would like to know how to interpret it more intuitively. I understand the definitions of inf and sup, however not so much what the book is trying to tell me here:
Let $g_i : \mathbb{R}^n \to \mathbb{R} \cup \{+\infty \}$ for $i \in I$ be a family of convex functions. For every $x \in \mathbb{R}^n$ define the function $g: \mathbb{R}^n \to \mathbb{R} \cup \{ + \infty\}$ as $$g(x) := \sup\limits_{i\in I}g_i(x)$$
Now this is for a convex optimization course, but basically I understand the definition of supremum to be the minimum element of a set of the upperbound.
My interpretation is that $g_i$s are a family of convex functions.
what exactly is $i \in I$ ? And what exactly is the interpretation of our $g(x):= (…) $ with the supremum, $i \in I$ etc ?
Thanks in advance for any help
Best Answer
In this case $I$ is an index set. For example, you could take $I = \{1,2,3\}$, $I = \mathbb{N}$, or even $I = \mathbb{R}$.
Your set of functions is then going to be $\{g_1, g_2, g_3\}$, $\{g_1,g_2, g_3, g_4, \dotsc\}$, or $\{g_i: i\in\mathbb{R}\}$ respectively (unfortunately there is no better way to write the last set since we can't count off real numbers).
From here, we define a new function $g$ as follows: fix an $x\in\mathbb{R}^n$. Compute all the $g_i$'s for this value of $x$ (for example, we may have $g_1(x) = 1$, $g_2(x) = 2$ and $g_3(x) = 3$). Then we take supremum of all these values, and let $g(x)$ denote that value (in this case, $3$). We then move on to another $x\in\mathbb{R}^n$, and in this manner we pick values of $g$ for all of $\mathbb{R}^n$. We need to take the supremum in our definition because the set of $g_i$'s evaluated at $x$ may be infinite. Like for the second $I$, I could have $\{g_i(x): i\in\mathbb{N}\} = \{0.9, 0.99, 0.999, 0.9999,\dotsc\}$. Here, we let $g(x) = 1$.
Edit: One more thing, since there is an important case I didn't address. I could also have my $g_i$'s infinitely growing, like $\{g_i(x): i\in\mathbb{N}\} = \{100, 200, 300, 400, \dotsc\}$. Based off the definition, we assign $g(x)$ the value $\infty$ if this happens.