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How do you convert $\min |x| + |y|$ to a linear program?

Is this method correct?

$$\min w + z$$
$$w >= x$$
$$w >= -x$$
$$z >= y$$
$$z >= -y$$

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Yes, this is a correct linearization. An alternative approach introduces two nonnegative values and one equality constraint for each absolute value. See this post for both approaches.

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