For this question $i\in\{1,2,\dots k\} $
Given samples $$ x_{i}\in\mathbb{R}^{8},y_{i}\in\mathbb{R} $$ Assume a linear model such that $y_{i}=a^{T}x_{i}-b+\varepsilon_{i} $ where $\varepsilon_i$ is some random gaussian noise. Assume a minimum error criterion $$p^{*}=\min_{a\in\mathbb{R}^{8},b\in\mathbb{R}}\max_{i\in\left\{ 1,2,\dots k\right\} }\left|y_{i}-\left(a^{T}x_{i}-b\right)\right| $$
This leads me to $$ r_{9\times1}=\left[\begin{array}{c} \boldsymbol{a}\\ \hline b \end{array}\right],\tilde{x_{i_{9\times1}}}=\left[\begin{array}{c} x_{i}\\ \hline -1 \end{array}\right] $$ $$ r^{T}x_{i}'=a^{T}x_{i}-b\Rightarrow y_{i}-r^{T}x_{i}'=y_{i}-a^{T}x_{i}-b $$ $$ \min_{r\in\mathbb{R}^{9}}\max_{i\in\left\{ 1,2,\dots k\right\} }\left|r^{T}x_{i}-y_{i}\right|$$ Using linprog, how do i find $p^*$ and $a,b$ ? I tried using the common reduction in here https://math.stackexchange.com/a/2591600/841573 $$ \min t $$ $$ s.t:-t\le r^{T}x_{i}^{'}-y_{i}\le t\iff\begin{cases} r^{T}x_{i}'-y_{i}\le t\\ -\left(r^{T}x_{i}'-y_{i}\right)\le t \end{cases}$$ How can I now find $r^T$ and $t$ where $t$ should be the minimal error and $r^T$ should be an estimation of $a,b$? For numpy\matlab I need to form it as a linear program but the input parameters are constant , how can i expect $t$ as a solution while i give it $t$ as a parameter?