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I'm looking for a way to write Chebyshev approximation in standard linear programming form,

Let's say my Chebyshev approximation have the following for:

$ min_{p,t} t $
s.t.
$ -t1_T \le h - Fp \le t1_T $

when $ t\in\mathbb{R}; 1_T=\begin{bmatrix}1\\\vdots\\1\end{bmatrix}; h\in\mathbb{R}^T; p\in\mathbb{R}^n; F\in\mathbb{R}^{T\times n}$

I'm trying to write it the following form:

$ min_{x} c^T x $
s.t.
$ Ax \le b $

when $ x,b,c $ are vectors and $ A$ is a matrix

my steps are:

$ min_{p,t} t $
s.t.
$ -t1_T \le h - Fp \land h - Fp \le t1_T $

$ min_{p,t} t $
s.t.
$ Fp - h \le t1_T \land h - Fp \le t1_T $

$ min_{p,t} t $
s.t.
$ \begin{bmatrix}Fp - h\\h - Fp\end{bmatrix} \le t\begin{bmatrix}1_T\\1_T\end{bmatrix} $

$ min_{p} \begin{bmatrix}Fp - h\\h - Fp\end{bmatrix} \le \begin{bmatrix}1_T\\1_T\end{bmatrix} $

so: $$ c^T = \begin{bmatrix}1_T\\1_T\end{bmatrix}; x = \begin{bmatrix}p\\p\end{bmatrix}; b = \begin{bmatrix}-h\\h\end{bmatrix}; A = \begin{bmatrix}-F\\F\end{bmatrix} $$

Is it right?

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