Linear Programming: Purely linear formulation (LP) for the expression "if x > 0; y =0 / if y > 0; x=0" without binary variables

Question

I have a problem which often occurs in energy system modelling (modelling of storages) where charging of a storage system is not allowed to when it is discharging and vice versa.

Let x and y be continous positive decision variables that represent the charging/discharging power of the storage unit. Now I am trying to express the following logic in equations or inequalities:

if x > 0:
    y = 0
if y > 0:
    x = 0

I know this can be realized with binary variables via the following equations if b is a binary variable and x_ub and y_ub represent the respective upper bound of the variables:

x <= b * x_ub
y <= (1-b) * y_ub

This results in a mixed-integer linear programm (MILP) and of course induces all the properties of a MILP which lay in higher runtimes, other interpretations of the solution, ...

Another way to get a pure LP formulation is to work with (very low) costs that are attached to x and y so that only one of both variables is set to a positive value. But this also affects my objective, has other disadvantages as is is somewhat of a hotfix.

So here's my question: Is there any way to express the logic above (if/else) in a purely linear program (LP)?

Thanks in advance!

Answer 1

This can't be expressed as an LP. Conceptually you're after the following constraints: $x \geq 0, y \geq 0, xy=0$.

The third constraint isn't convex, so there's no way you could represent that in terms of LP constraints, since those can only define convex regions.