If-then constraint with continuous variables

Question

I was usually using if-then constraints with integer variables but ended up using continuous variables and got confused. I have variables $x_{ij}\in\mathbb{R}_{\geq 0}$, and would like to force the program: if $x_{ij}>0$, then $x_{ji}=0$, where $i,j\in\mathcal{I}$. When I use the following (in which $\mathbb{M}$ is a big number), it looks like the constraint serves to the purpose.

$$x_{ij}\geq \mathbb{M}x_{ji} \qquad \forall i,j\in\mathcal{I}.$$

However, it does a bit further than what is expected and undesired, which is $x_{ji}=0$, when $x_{ij}=0$. The symmetry of this will be conflicting. My variable essentially denotes a flow amount from node $i$ to node $j$, and I would like to ensure: if flow occurs from $i$ to $j$, it should not occur from $j$ to $i$.

Answer 1

Introduce a binary variable $y_{i,j}$ and linear constraints \begin{align} x_{i,j} &\le M y_{i,j} \tag1 \\ y_{i,j} + y_{j,i} &\le 1 \tag2 \end{align} Constraint $(1)$ enforces $x_{i,j} > 0 \implies y_{i,j} = 1$. Constraint $(2)$ enforces $y_{i,j} = 1 \implies y_{j,i} = 0$. Constraint $(1)$ (with the roles of $i$ and $j$ interchanged) enforces $y_{j,i} = 0 \implies x_{j,i} = 0$.

Answer 2

For sake of completeness: An alternative to using an additional binary variable and a big M would be to use an SOS constraint. An SOS constraint specifies a set of variables, at most one of which may be non-zero.