# Linear optimization problem.

Given a $m$ x $n$ matrix $A$, $m$-vectors $b$ and $y$, and $n$-vectors $c$ and $x$.
Write the dual $LP$ problems $P$ and $P^d$ in the standard form.

Whether $x$ (respectively, $y$) is a feasible vector for $P$ (respectively, for $P^d$)?
Whether x (respectively, y) is an optimal solution for $P$ (respectively, for $P^d$)?
Whether the complementary slackness conditions hold for $P$ (respectively, for $P^d$)?

Consider the cases a-g listed below and explain your answers in each case.

a.) $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $b^T = (8,18)$, $c^T = (2,1)$, $x^T = (6,0)$, $y^T = (0,2/3)$

b.) $A = \begin{bmatrix} 1 & 1 & 2 \\ 2 & 3 & 4 \\ 7 & 6 & 2 \end{bmatrix}$, $b^T = (2,3,8)$, $c^T = (8,9,4)$, $x^T = (6/9,5/9,0)$, $y^T = (0,5/3,2/3)$

c.) $A = \begin{bmatrix} 1 & 1 & 2 \\ 2 & 3 & 4 \\ 6 & 6 & 2 \end{bmatrix}$, $b^T = (2,3,8)$, $c^T = (8,9,5)$, $x^T = (1,1/3,0)$, $y^T = (1,1,1)$

d.) $A = \begin{bmatrix} 3 & 2 \\ 1 & -2 \end{bmatrix}$, $b^T = (6,1)$, $c^T = (2,1)$, $x^T = (1,2)$, $y^T = (1,1)$

e.) $A = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 2 & 1 & -1 & -1 \\ 0 & -1 & 0 & 1 \end{bmatrix}$, $b^T = (40,-5,10)$, $c^T = (1,-3,1,4)$, $x^T = (35/3,0,55/3,10)$, $y^T = (2,2,0)$

f.) $A = \begin{bmatrix} 2 & -6 & 2 & 7 & 3 & 8 \\ -3 & -1 & 4 & -3 & 1 & 2 \\ 8 & -3 & 5& -2 & 0 & 2 \\ 4 & 0 & 8 & 7 & -1 & 3 & \\ 5 & 2 & -3 & 6 & -2 & -1\end{bmatrix}$, $b^T = (1,-2,4,1,5)$, $c^T = (18,-7,12,5,0,8)$, $x^T = (2,4,0,0,7,0)$, $y^T = (1/3,0,5/3,1,0)$