Let $G=(V,E)$ be a graph. I would like to identify an eulerian cycle in $G$ with minimum cost, with an integer programing approach:
$x_{ij}$ are integer variables that denote the number of times that edge $(i,j)$ is used.
$$ \min \; \sum_{(i,j)\in E}c_{ij}x_{ij} $$ subject to: \begin{align} \sum_{i,(i,j)\in E}x_{ij}&=\sum_{i,(j,i)\in E}x_{ji} \quad \forall j \in V \tag{1}\\ x_{ij} &\ge 1 \quad \forall (i,j)\in E \tag{2} \end{align}
Is this a valid formulation? Can I relax the variables to be continuous?