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Suppose that I have a system of linear equations for a variable $x\in\mathbb{R}^n$:

$$a_i^Tx\ge b_i,\quad i\in I,\quad (1)$$

where $a_i\in\mathbb{R}^n$ and $b_i\in\mathbb{R}$ for all $i\in I$, and $I$ is a finite set of index.

Then, by definition, the characteristic cone of $(1)$ is given by

$$K=\text{cone}\left\{ \begin{pmatrix} a_i\\b_i \end{pmatrix}\in\mathbb{R}^{n+1}\ \Bigg|\ i\in I \right\}=\left\{v\in\mathbb{R}^{n+1}\ \Bigg|\ v=\sum_{i\in I}\alpha_i \begin{pmatrix} a_i\\b_i \end{pmatrix},\ \alpha_i\ge0\ \forall i\in I \right\}$$

If $I$ is an infinite set if index, how is the characteristic cone $K$ defined?

I think it can be defined similarly, constraining the sum $v=\sum_{i\in I}\alpha_i \begin{pmatrix} a_i\\b_i \end{pmatrix}$ to be over a finite subset of $I$, but then, the cone will not be closed, which can lead to different characteristic cones for equivalent systems of linear inequalities.

Also, if someone know any textbook or webpage where this definition can be found, please help me with the reference.

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Intuitively the sum will just be over an infinite set, this is in the case that $I$ is countable.

If $I$ is uncountable, you would integrate the coefficients.

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