Let following linear problems:
Primal Problem: \begin{eqnarray*} \textrm{Min}\quad c^T x & & \\ \textrm{s.t.}\quad Ax & \geq & b \end{eqnarray*} and its dual problem: \begin{eqnarray*} \textrm{Max}\quad b^T \lambda & & \\ \textrm{s.t.}\quad A^T\lambda & = & c \\ \lambda & \geq & 0 \end{eqnarray*}
and suppose that (P) and (D) are both feasible. Show that there exist optimal points $X^\star$ and $\lambda ^\star$ such that:
\begin{array}{l} \textrm{1)}\quad AX^\star \geq b \\ \textrm{2)}\quad A^T\lambda^\star = c \,\, , \,\, \lambda^\star \geq 0 \\ \textrm{3)}\quad (AX^\star-b)\bot \lambda^\star \\ \textrm{4)}\quad AX^\star-b+\lambda^\star >0 \end{array}
