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For an LP problem where $x_1,\dots,x_n$ are free variables (which may take positive or negative values), I want to bound the sums of $a_i\cdot x_i$ where $x_i>0$, and where $x_i<0$.
I suspect this requires MIP with a sign variable for each $x_i$.
Any pointers or suggestions would be appreciated.
This can be done using the standard trick of splitting each free variable into the difference of two nonnegative variables:
$$x_{i}=u_{i}-v_{i}$$
where $u \geq 0$ and $v \geq 0$.
If your constraint is
$$\sum_{i=1}^{n} a_{i} \max(x_{i},0) \leq b$$
with $a \geq 0$, then this can be written as
$$\sum_{i=1}^{n} a_{i} u_{i} \leq b.$$
For the constraint
$$\sum_{i=1}^{n} a_{i} \min(x_{i},0) \geq -b$$
with $a \geq 0$, use
$$\sum_{i=1}^{n} a_{i} v_{i} \leq b.$$
Note that it's easy to formulate related constraints that can't be formulated using LP. For example, a constraint like
$$\sum_{i} a_{i} \max(x_{i},0) \geq b$$
with $a_{i} \geq 0$ is nonconvex and thus can't possibly be represented with LP. In that case, you might formulate the problem using 0-1 integer variables to encode whether variables are positive or negative.
