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Im tring to transform the following minimization problem:

$$\arg \min_{x \in \mathbb{R}^d : Wx=y} \left\|x\right\|_1 $$

into a linear programming problem which is a program with the following form:

$$\max_{w \in \mathbb{R}^d} \langle u,w \rangle $$

subject to

$$ Aw \ge y$$

the linearprog receives:

$$\text{linearprog}(A,u,v)$$

I've already found the first constrain which is :

$ Wx = y $ using the $A=\binom{W}{-W} , v=\binom{y}{-y}$ which gives $Wx \ge y, Wx \le y $

I'm still having hard time to fit the constraints to mininize $L_1$ norm ($\left\|x\right\|_1$).

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    Maybe I misunderstand your notation, but you seem to have a nonlinear problem. Why do you think that you can cast that into a linear program? – 2017-01-18

1 Answers 1

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$$\min \left\| x\right\|_1$$

is equivalent to

$$\min \sum_i^n \max(x_i, -x_i)$$ which is equivalent to

$$\min \sum_i^n p_i$$

subject to

$$x_i \leq p_i$$ $$-x_i \leq p_i$$

Edit:

For your original problem, the linear programming formulation is

$$\max \langle \begin{bmatrix} 0 & \cdots & 0 & -1 & \cdots & -1\end{bmatrix}^T, \begin{bmatrix} x & p\end{bmatrix}^T \rangle $$

subject to

$$ \begin{bmatrix} -I_n & I_n \\ I_n & I_n \\W & 0 \\ -W & 0\end{bmatrix} \begin{bmatrix} x \\ p\end{bmatrix}\geq \begin{bmatrix} 0 \\ 0 \\ y \\ -y\end{bmatrix}$$

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    Thanks, but eventualy i need to convert the minimization problem into a linear programming problem, and i still don't see how can you convert this constrain into a linear programming problem in matlab (how do you find the appropriate A,u,v)? – 2017-01-18
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    edited my solution, can you see the corresponding component? – 2017-01-18
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    Thank you, that was very helpful. – 2017-01-18