Many thanks! But I tried this and get status is INFEASIBLE.
If l equals -2, then 0<= neg <=-2? This seems not to be straightforward to me. Also, in the link
https://docs.mosek.com/modeling-cookbook/mio.html#exact-absolute-value
you posted, there seems should be
@constraint(model, pos <= M * z)
@constraint(model, neg <= M * (1 - z))
and the constant M is an a priori known upper bound on |x|, which should be positive.
And indeed in my case the objective to be optimized is of type AffExpr, so I added a line
@constraint(model, x == x_AffExpr)
where x_AffExpr is a known expression.
It would be great if you can explain a little bit more. Thanks!