Dual of a linear program.

Question

I have a linear program of a set covering problem; \begin{align} & &\underset{}{\text{min}} &\sum_{i=1}^n c_ix_i & & & \\ & \text{s.t.} & & \sum_{i=1}^n b_{i,j}x_i \geq d_j & \forall j=1,\dots,m\\ & & & x_i \geq 0 & \forall i=1,\dots,n \end{align} and I want to write the dual of this primal by rewriting it to the dual of a linear program in standard form and then taking the dual, because the dual of the dual is the primal linear program (right?). I finally achieve, \begin{align} & &\underset{}{\text{max}} &\sum_{j=1}^m d_jy_j & & & \\ & \text{s.t.} & & \sum_{j=1}^m b_{i,j}y_j \leq c_i & \forall i =1,\dots,m\\ & & & y_j \geq 0 & \forall j=1,\dots,m \end{align} Is this correct?

Answer 1

Looks fine, just a minor typo.\begin{align} & &\underset{}{\text{max}} &\sum_{j=1}^m d_jy_j & & & \\ & \text{s.t.} & & \sum_{j=1}^m b_{i,j}y_j \leq c_i & \forall i =1,\dots,\color{blue}n\\ & & & y_j \geq 0 & \forall j=1,\dots,m \end{align}