I need to define the dual of the assignment-like problem where the cost function is defined as the Hadamard product of two matrices $C=[c_{ij}]$ and $X=[x_{ij}]$ as follows: \begin{align} \text{minimize $\sum\limits_{i=1}^n\sum\limits_{j=1}^n c_{ij}x_{ij}$}& \\ \sum_{j=1}^n x_{ij} &= 1 &&\forall i \\ \sum_{i=1}^n x_{ij}r_{i} &\le b_j &&\forall j \\ x_{i,j} &\in \left\{0,1\right\} &&\forall i,j \end{align}
I tried to construct the dual problem from this primal formulation but I am not sure how to define it. Anyone can help ?
If you want the dual of the linear programming relaxation of the primal, I suggest omitting the upper bound on $x_{i,j}$ because it complicates the dual and is already implied by nonnegativity and the first constraint. Introduce a (free) dual variable $\alpha_i$ for the first constraint and a nonpositive dual variable $\beta_j$ for the second constraint. The primal is minimization, so the dual is maximization. The primal objective coefficients become the dual RHS, and vice versa. The dual constraint matrix is the transpose of the primal constraint matrix.