Trouble understanding a passage in Nonlinear Programming by Bertsekas

Question

I am reading Nonlinear Programming by Bertsekas, and the chapter on duality starts like this: we define the primal problem as $$\begin{align*} &\min f(x)\\ &x \in X\\ &g(x) \le 0 \end{align*}$$ where $X \subseteq \mathbb{R}^n$ and $g: \mathbb{R}^n \to \mathbb{R}^m$. Then the author uses $f^*$ to denote the solution of this problem, using $\inf$ now: $$\begin{align*} &\inf f(x)\\ &x \in X\\ &g(x) \le 0 \end{align*}$$

Then the author goes on to say:

Note that the definition of $f$ and $g_i$ [the components of $g$] is immaterial outside $X$, so if in a given problem the cost function and/or some of the constraints are defined over a domain $D \subset \mathbb{R^n}$, we can introduce $D$ as part of the set $X$, and redefine these functions arbitrarily outside $D$. Unless the opposite is clearly stated, we will assume throughout this chapter the following:

Assumption 6.1.1: (Feasibility and Boundedness) There exists at least one feasible solution for the primal problem and the cost is bounded below, i.e. $- \infty < f^* < \infty.$

There are several things I don't understand here:

$1)$ Why switch from the minimum to the infimum?

$2)$ Why would we ever consider the feasible region as defined by an abstract set $X$, AND by inequalities? If we allow ourselves to use abstract sets, why don't we just cobine all the constrains into one set?

$3)$ What does it mean to "introduce $D$ as part of $X$"? I assume $X$ must already be contained in $D$, so what does it mean to make $D$ "a part" of $X$? My best guess is that we redefine $X$ as $D$. But why would we modify our domain of feasibility? We are making up a new problem, whose solution may not be the same as the original problem, in whose solution we are interested.

$4)$ Is Assumption 6.1.1 in any way connected to the discussion about $D$ above it? I don't see how it would be, but it's right below the discussion about $D$.

Thank you very much.

Answer 1

1. Switching from minimum to infimum allows you to discuss problems where the objective value is bounded below but a minimum is never attained (such as "minimizing" $f(x)=1/x$ over the domain $x>0$).
2. It is common to state domain limits ($x\ge 0$, $x \le 5$, $x$ integer) separately from "functional constraints" ($x_1^2 + x_2^2 - 1 \le 0$ etc.). You may want to assign Lagrange multipliers to the latter, take derivatives, or do other stuff than would not necessarily make sense for the domain limits.
3. This is a bit opaque, but my best guess is that the author intends $D$ and $X$ to intersect without either necessarily being contained in the other. Off hand, I can't think of a suitable example. If $D \subset X$, it certainly does make sense to just change $X$ to $D$ in the problem statement.
4. I don't think the Assumption is explicitly tied to the confusing passage, although it is connected in the sense that the assumption implies $D\cap X \neq \emptyset$.