I'm doing the Convex Opt. mooc taught by Stephen Boyd offered by Standford Lagunita. This question is being asked:
Consider the cone: $\ K= \{(x_1,x_2) \ | \ 0 \le\ x_1 \le\ x_2 \} \subseteq {\mathbf{R}}^2$. Which of the following are true?
And then we have 2 options:
1) $ \ (1,3)\le _K(3,4) $
The definition of generalized inequality says: $$ x \le _Ky $$
$$ y-x\in K $$
Which leave us: $(3,4)-(1,3)=(2,1)$, which is false since the restriction for our cone says that $x_1 \le\ x_2$
No issues in there. But in the next one I don't know what should be the right procedure, therefore I don't know how to answer.
2) $(-1,2) \ge _{K^*}0$
The definitions for generalized inequealities are all in the form of $ x \le y $, how should I proceed? Can I flip the terms $x$ and $y$ and say this?
$$x-y \in K^*$$
Is it wrong to assume this? If not, I'd have to define what is $K^*$ given $K$, right? the restrictions stay the same? how can I do that?
Sorry, I know this is simple for a lot of you, but I need some help to figure this out. Thanks!