Gurobi add constraint which two variables cant be zero at the same time

Question

I'm currently using Gurobi in python and trying to add a constraint that the variable will not equal to 0 at the same time, say a != 0 or b != 0, both a and b can be negative or positive. But I can't find a proper way to add this constraint. Does anyone have idea about this?

Edit: The problem I'm trying to solve is to find the proper plane that has the smallest distances in summation for a list of points. The equation I used is $\sum_{i=0}^n \frac{|Ax_{i} + By_{i} + Cz_{i} + D|}{\sqrt{A^2 + B^2 + C^2}}$ where $x_{i}, y_{i}, z_{i}$ refer to the point, A,B,C,D refer to the plane equation Ax + By + Cz + D = 0, and thus A,B,C cant be 0 at the same time.

Answer 1

There's a trick I sometimes use in situations like this. First, you need to confirm that for any optimal solution $(A, B, C, D)$ and any scalar $\lambda \neq 0$, $\lambda\cdot (A, B, C, D)$ is also optimal. It obviously selects the same plane, so chances are good this will hold.

Assuming it does, generate a random vector $\alpha$ distributed uniformly over $[-1, 1]^3$ and add the constraint $\alpha_1 A + \alpha_2 B + \alpha_3 C = 1$ to the model. This obviously rules out $A = B = C = 0$. For any optimal $(A, B, C)$ for which the left side is not zero, we can by assumption multiply by a constant to get another optimal solution that sums to 1. So the only potential issue is whether the optimal solution might have inner product 0 with $\alpha$. Since we chose $\alpha$ using a continuous distribution over a region with positive volume, that has probability zero of occurring. So unless you are living under a curse, this ought to work.