Dual linear program

For a primal maximization problem $Ax\le b$, $x\ge 0$, the dual minimizes $b^{T}y$ subject to $A^{T}y\ge c$, $y\ge 0$.

A dual variable is a multiplier associated with a primal constraint in a dual linear program. In a minimization linear program with equality constraints $Ax=b$ and nonnegative primal variables, the Lagrangian $c^{T}x+y^T(b-Ax)$ yields the dual objective $b^{T}y$ and inequalities $A^{T}y\le c$; the equality-constraint multipliers $y$ are unrestricted in sign. For a transportation problem, row and column dual variables give potentials $u_i,v_j$ satisfying $u_i+v_j\le c_{ij}$. Complementary slackness makes occupied routes attain equality.