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I have a linear program that I am trying to rewrite in a (possibly not) simpler form, most likely with a 1-norm constraint. The problem is quite simply

$\displaystyle\min_{x}{\sum_{i}{x_i}} \quad\text{ subject to }\quad -y_i \leq x_i \leq y_i, \ \forall i$

The aim is to remove the extra variables $x_i$ and reduce to a single optimisation problem (convex or not) in $y_i$ only. Any ideas?

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Because $x_i\ge -y_i \forall i$ we have $$\min_x\sum_{i}{x_i} = -\sum_{i}{y_i},$$ attained when $x_i=-y_i\forall i.$

The variables $x_i$ are eliminated as you wish.

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