I'm working on the following problem and am unsure if I'm doing it right:
Let $y = (y_1,y_2,....,y_N)^T$ be an $N$- vector of non-negative real numbers. Consider the following problem:
$$min_{x \in \mathbb{R^N}} \sum_{i=1}^N |x_i - y_i| + a*\sum_{i=1}^{N-1}|x_{i+1} - x_i|$$ with $x \ge 0$ and $a > 0$.
Bring this problem into the Standard Form, i.e:
$min_{x \in \mathbb{R^n}} c^T*x $ with $Ax=b$ and $x \ge 0$.
I proved earlier that for any real number $x \in \mathbb{R}$ holds:
$|x| = inf_{(y_1,y_2) \in \mathbb{R^2}} \ y_1 + y_2 \ $ such that $x = y_1 - y_2 , \ y_1 \ge 0 , \ y_2 \ge 0$
With that in mind we can rewrite the above problem to:
$$min_{x \in \mathbb{R^N}} \sum_{i=1}^N |x_i - y_i| + a*\sum_{i=1}^{N-1}|x_{i+1} - x_i|$$
$$= min_{x \in \mathbb{R^N}}\sum_{i=1}^N inf_{x_i+y_i \ = \ x_i-y_i } \ x_i + y_i \ \ + \ \ a*\sum_{i=1}^{N-1} inf_{x_{i+1}+x_i \ = \ x_{i+1}-x_i} \ x_{i+1} + x_i $$
$$= min_{x \in \mathbb{R^N}}\sum_{i=1}^N x_i + y_i \ \ + \ \ a*\sum_{i=1}^{N-1} x_{i+1} + x_i $$ such that $x_{i+1}+x_i \ = \ x_{i+1}-x_i$ and $x_i+y_i \ = \ x_i-y_i$
$$= min_{x \in \mathbb{R^N}}\sum_{i=1}^N (a+1)*x_i + y_i $$ such that $x_{i+1}+x_i \ = \ x_{i+1}-x_i$ and $x_i+y_i \ = \ x_i-y_i$