[Scip] why the dual solution is non-positive?

[Tobias Achterberg]

XiangyongLi wrote:
> min \sum_{(i,j)} c_{ij}x_{ij}
> s.t. Az ==1 for each k (1)
> Dz - x_{ij} <= 0, for each arc (i,j) and each k (2)
> x, z >=0
> For constraints (2), I define dual multiplier v_{ijk}. So in my dual
> model, I should have \sum_{k} v_{ijk} <= c_{ij}. And dual multiplier
> v_{ijk} should be non-negative. But the SCIP returns non-positive v_{ijk}.

Not correct. Note that the coefficients of the x_{ij} variables in the matrix is -1. 
Therefore, you have the dual constraint

    \sum_{k} -v_{ijk} <= c_{ij},

and because the primal inequality is a '<=' row of a minimization problem, the dual 
multiplier v_{ijk} must be non-positive.


Tobias