If i have a standard LP problem:
$$\min \mathbf{d}^T \mathbf{x}$$
subject to
$$\mathbf{B}\mathbf{x}=\mathbf{f},\qquad \mathbf{x} \geq 0$$
$\mathbf{y}$ is the optimal solution and $\mathbf{z}$ is the optimal solution to the dual problem
Now, for the same cost function $\mathbf{d}$, if $\mathbf{f}$ is replaced by $\mathbf{b}$ then $\mathbf{x}$ becomes the new optimal solution.
How can it be shown:
$$\mathbf{z}^T (\mathbf{b}-\mathbf{f}) \leq \mathbf{d}^T (\mathbf{x}-\mathbf{y})$$