I would like to find the optimal set $ \{ x_i \} $ given $ L $ and $ \{ a_i \} $ that minimizes the problem below. My first thought was to use linear programming. Is there a transformation that makes it possible, or do I need a more general optimization technique?

$$ \min_{x_i} \left[ 2\sum_i | x_i | + \max_i | x_i + a_i | \right]$$ $$ \mathrm{s.t.} \sum_i (x_i + a_i) \le L$$


You haven't told us what set $i$ ranges over, so I'll just assume $i=1, 2, \ldots, n$.

A standard trick in LP formulation of problems with absolute values is to introduce auxiliary variables and constraints with the basic idea that

$\min | x | $

is equivalent to

$\min t $

$t \geq x $

$t \geq -x $

Applying that idea to your problem, introduce auxiliary variables $t_{i}$, $i=1, 2, \ldots n$, and $s$. Then formulate the problem as:

$\min 2 \sum_{i=1}^{n} t_{i} + s $

subject to

$t_{i} \geq x_{i}$, $i=1, 2, \ldots n$.

$t_{i} \geq -x_{i}$, $i=1, 2, \ldots n$.

$s \geq x_{i}+a_{i} $, $i=1, 2, \ldots, n$.

$s \geq -(x_{i}+a_{i}) $, $i=1, 2, \ldots, n$.

$\sum_{i=1}^{n} (x_{i}+a_{i}) \leq L$

  • $\begingroup$ Thanks, but what happened to the $\max$? $\endgroup$
    – user2303
    Jul 2 '13 at 20:29
  • 3
    $\begingroup$ $s$ is larger than any of the $x_i+a_i$ and $-(x_i+a_i)$, so it is larger than their max. What Brian has shown you is a standard trick to convert a problem like yours into a linear problem with linear constraints -- the sort of problem for which there are many standard implementations. $\endgroup$ Jul 2 '13 at 21:30
  • $\begingroup$ As Wolfgang explained, $s$ will be greater than or equal to $ | x_{i}+a_{i} | $ for $i=1, 2, \ldots, n$. Also, since $s$ is being minimized in the objective, it will come out equal to the maximum of the absolute values of $x_{i}+a_{i}$. $\endgroup$ Jul 3 '13 at 3:43

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