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Is there any method to linearize the following optimization problem?

\begin{align} min_{x,y} &~~ c~[x; y] \\ st &~~ \sum x\leq \alpha_1 \\ &~~ \sum |y|\leq \alpha_2 \\ &~~ \sum y= 0 \\ &~~ x+|y| \leq 1 \\ &~~ (x,y)\in \{0,1\}*\{-1,0,1\} \end{align}

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    $\begingroup$ Have you tried using the standard trick for absolute values in convex inequalities? That is, replace $|z| \leq b$ by the pair of constraints $z \leq b$ and $-z \leq b$. You can then turn your problem into a mixed integer linear programming problem. $\endgroup$
    – Brian Borchers
    Jan 30, 2022 at 0:29

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I don’t have any to mind, but you can use a family of functions which are parameterized and smooth with absolute value as their limit. Then you can solve for multiple, usually shrinking, values of the parameter until you’re happy with the level of minimization.

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