I would like to convert the following condition for $x$ \begin{align} x = N \lambda, \lambda \geq 0 \end{align} to a pure linear inequality form, i.e. find an $L$ and eliminate $\lambda$ \begin{align} L x \leq 0 \end{align} The first condition basically means "$x$ is in the cone generated by the columns of $N$", hence the set $x$ lives in should be a polytope with all faces containing the origin, justifying the form $Lx \leq 0$. But how exactly can I express $L$ in terms of $N$?
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$\begingroup$ In my case, $N$ is full column rank. Can I do $N^\top x = N^\top N \lambda$, solve for $\lambda = (N^\top N)^{-1}N^\top x \geq 0$ ? $\endgroup$– Jacob DiCommented Aug 16, 2019 at 20:31
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$\begingroup$ No, you can't. What that formula computes is the projection of $x$ over the range of $N$. For instance, if $N=[1, 0]^T$, then $x=[1,1]^T$ satisfies the inequality in your comment, but not the one in your question. $\endgroup$– Federico PoloniCommented Aug 16, 2019 at 20:41
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$\begingroup$ I see now. You are right. $\endgroup$– Jacob DiCommented Aug 16, 2019 at 21:37
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