# Trouble understanding a passage in Nonlinear Programming by Bertsekas

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.

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 $$D\cap X \neq \emptyset$$.
• Thank you so much for your answer! In regards to $2$: I see how "x is an integer" is a different type of constraint, but I thought that $x \ge 0$ or $x \le 5$ would always be considered "functional constraints". – Ovi Oct 3 '20 at 0:57