I have a given formulation that looks like this (just for the constraints):
$$\sum_i \beta_{i,j} \geq \alpha_j,\qquad\forall j$$ $$\alpha_j \geq \sum_i f_{i,j},\qquad\forall j$$ $$\alpha_j \geq 0, f_{i,j} \geq 0$$
In order to reduce the size of my formulation, I eliminated the variable $\alpha_j$ by the means of a valid inequality (may I call it a projection?):
$$\sum_i \beta_{i,j} \geq \sum_i f_{i,j},\qquad\forall j$$ $$f_{i,j} \geq 0$$
However, I am now interested in the dual values for each of these constraints (actually, just $\sum_i \beta_{i,j} \geq \alpha_j$). How these dual values are related to the ones of the reformulation (mostly $\sum_i \beta_{i,j} \geq \sum_i f_{i,j}$)?