Let $f: \mathbb{R^n} \rightarrow \mathbb{R}$ and $g: \mathbb{R^n} \rightarrow \mathbb{R^m}$ and $C \subset \mathbb{R^n}$. And further suppose we know that $$\alpha = \inf[f(x):g(x) \leq 0, x \in C]$$ The inequality on $g(x)$ just means that each component is non-positive. Construct the follow function to examine what happens when the constraints are changed: $$p(y) = \inf[f(x):g(x) \leq y, x \in C]$$ The book I'm reading states:
Obviously $p(y) \leq \alpha$ if $y \geq 0$ and $p(y) \geq \alpha$ if $y \leq 0$
Alas, I don't see it -- how do I show this to myself?
The quick answer to your question is that if $y\ge 0$, then you're taking the inf over a bigger set, so that inf has to be smaller. If $y\le 0$, then you're taking the inf over a smaller set, so that inf has to be bigger.
In more detail, if $y \ge 0$, $$ \{f(x) : g(x) \le y\} = \{f(x) : g(x) \le 0\} \cup \{f(x) : 0 < g(x) \le y\}\supset \{f(x) : g(x) \le 0\}, $$ so that $p(y) \le p(0)$, and similarly if $y \le 0$, $$ \{f(x) : g(x) \le y\} = \{f(x) : g(x) \le 0\} \setminus \{f(x) : y < g(x) \le 0\} \subset \{f(x) : g(x) \le 0\}, $$ so that $p(y) \ge p(0)$.