Operations research : minimize the absolute value instead of square

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I am working on an operations research problem In a previous work, they minimized the square of a-b : min sum (a-b)² I would like to work on a linear problem, by minimizing the absolute value of (a-b) I do not how how to provide a mathematical proof Knowing that it does not change the purpose of the problem, and in the previous work, the use of square was not justified. Thank for the support