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Let $$\mathbb{R}_+^n = \{x = (x_1,...,x_n)| x_1 \geq 0, x_2 \geq 0,\cdots, x_n \geq 0 \}$$ be the nonnegative orthant of $\mathbb{R}^n$.

How can we demonstrate the properties of the monotonic cone?

1) Let $c \in \mathbb{R}^n$ be given. Show that $c^Tx \geq 0 \forall x \in \mathbb{R}_+^n$ if and only if $c \in \mathbb{R}_+^n$.

2) Let $x, y \in \mathbb{R}_+^n$. Show that $x^Ty=0$ if and only if $x_iy_i = 0$ for each $i=1,...n$

First, I was thinking if we show the monotone cone is self-dual then $c \geq 0$, and therefore $c \in \mathbb{R}_+^n$.

Another part of it, suppose if $c \leq 0$, then $c$ not $\mathbb{R}_+^n$, and $c^Tx \leq 0$. But this is a contradiction.

user561304
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    This question looks like a homework exercise (I'm not saying that it is, only that it has that appearance). Since MSE is meant to be a repository of questions and answers (and not a solutions manual for mathematics exercises), your exercise is not really a good fit for the site. However, you could make your question fit in better by editing it to add some context. What attempt(s) have you made toward proofs? Where are you getting stuck? Are there any theorems that you are trying to invoke, or that you think might be helpful? – Xander Henderson May 12 '18 at 20:15
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    @XanderHenderson there is no need to invoke a theorem here... – Surb May 12 '18 at 20:28
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    @Surb I did not say that there was a need to invoke theorems; I asked the original poster to improve their question by providing some additional context. One possible way to provide such context it to indicate what tools they are trying to use (definitions, theorems, proving techniques, etc). If no such additional context is provided, then it is likely that this question will be closed, and ultimately deleted. – Xander Henderson May 12 '18 at 20:30
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    @XanderHenderson It is indeed (unfortunately) likely. On the other hand, it could also be that OP is reading the convex optimization book of Boyd and Vandenberghe for some reason other than homework and since the question is well formatted IMHO it shouldn't be closed. But well.... let's see – Surb May 12 '18 at 20:34
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    Try $n=1$ first. – Michal Adamaszek May 12 '18 at 20:39
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    @Surb tangentially, it is ultimately irrelevant whether it is homework. The presentation should have included more context, regardless to it being homework or not. "Looks like" should be understood in a literal sense, not as an insinuation that it is homework. Anyway, I'll stop here. Fortunately OP included some more information by now. – quid May 13 '18 at 00:19

1 Answers1

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Hints

1) One direction is obvious. For the other direction note that the canonical basis is nonnegative, that is $e^i \in \Bbb R_n^+$ for every $i$, where $(e^i)_j:=1$ if $i=j$ and $(e^i)_j:=0$ otherwise.

2) Again, one direction is obvious. For the other direction, suppose by contradiction that $x^\top y=0$ and $x_iy_i \neq 0$, then $x_iy_i>0$ as $x,y\in\Bbb R^n_+$ and so $x^\top y\geq x_iy_i>0$, a contradiction.

Surb
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  • on p.52 Boyds book they state the cone is self-dual. I just don't understand how I can use this fact. – user561304 May 12 '18 at 20:34
  • Well, then you asked to right question :). Probably you should add this comment in your question. Can you understand how to use my hints to prove your statements? – Surb May 12 '18 at 20:36