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# System prepended metadata

title: Untitled

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$\text{min}_{\underline{x}} \|y - A \underline{x} \|^2  +  c \cdot \sum_i \sum_{j \neq i} \left ( | x_i | - |x_j| \right )^2$ 
is equal to


$\text{min}_{\underline{x}, \underline{z}} \|y - A \underline{x} \|^2  +  c \cdot \sum_i \sum_{j \neq i} \left (  z_i  - z_j \right )^2 \text{s.t. } z_i = | x_i|$


which can be solved by iteratively solving:

$z^{k+1},x^{k+1} = \text{min}_{\underline{x},\underline{z}} \|y - A \underline{x} \|^2  +  c \cdot \sum_i \sum_{j \neq i} \left (  z_i - z_j \right )^2 +  \sum_i \left ( z_i - |x_i| - b^k \right )^2$

$b^{k+1}_i = b_i^k + |x^{k+1}_i| - z^{k+1}$

