# Difference between revisions of "Optimization with absolute values"

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Authors: Matthew Chan (mdc297), Yilian Yin (), Brian Amado (ba392), Peter (pmw99), Dewei Xiao (dx58) - SYSEN 5800 Fall 2020

Steward: Fengqi You

## Numerical Example

$\ min|x_{1}|+2|x_{2}|+|x_{3}|$ $\ s.t.x_{1}+x_{2}-x_{3}\leq 10$ $x_{1}-3x_{2}+2x_{3}=12$ We replace the absolute value quantities with a single variable:

$|x_{1}|=U_{1}$ $|x_{2}|=U_{2}$ $|x_{3}|=U_{3}$ We must introduce additional constraints to ensure we do not lose any information by doing this substitution:

$-U_{1}\leq x_{1}\leq U_{1}$ $-U_{2}\leq x_{2}\leq U_{2}$ $-U_{3}\leq x_{3}\leq U_{3}$ The problem has now been reformulated as a linear programming problem that can be solved normally:

$\min U_{1}+2U_{2}+U_{3}$ $\ s.t.x_{1}+x_{2}-x_{3}\leq 10$ $x_{1}-3x_{2}+2x_{3}=12$ $-U_{1}\leq x_{1}\leq U_{1}$ $-U_{2}\leq x_{2}\leq U_{2}$ $-U_{3}\leq x_{3}\leq U_{3}$ The optimum value for the objective function is $6$ , which occurs when $x_{1}=0$ and $x_{2}=0$ and $x_{3}=6$ .