Nocedal & Wright, example problem 13.1 I'm having a trouble understanding example problem 13.1 on page 371 of the 2nd edition of Nocedal & Wright:
\begin{aligned}
\begin{equation}
\min_x -4x_1 - 2x_2 \text{ s.t } \\
x_1 + x_2 + x_3 = 5 \\ 
2x_1 + \tfrac {x_2}{2} + x_4 = 8 \\
x \geq 0
\end{equation}
\end{aligned}
but ran into a calculation discrepancy described below that I thought might be preventing me from seeing how the update works in action:

NW assume that the initial basic basis is given by elements $\{3,4\}$
and the non-basic basis is given by elements $\{1,2\}$.
In NW's notation, they compute $$ s_N =  \begin{bmatrix} s_1 \\ s_2
 \end{bmatrix}
 = c_N - N^T \lambda =  \begin{bmatrix}
 -3 \\ -2 \end{bmatrix} $$
However, I get $\lambda = (B^T)^{-1}c_B = I(0,0)^T$ so that $$ s_N =
 c_N - N^T \lambda =  \begin{bmatrix}
 -4 \\ -2 \end{bmatrix}
 -  \begin{bmatrix} 1 & 2 \\ 1 & 1/2 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \end{bmatrix}
 = \begin{bmatrix}
 -4 \\ -2 \end{bmatrix}. $$

I checked the errata but didn't see anything listed, so I'm confused about what I'm missing.
 A: You are right, that does not make any sense. Another error in this example is that they claim that $\lambda=[-5/3; -2/3]$ found at the start of the third iteration is optimal, but it clearly is not since strong duality does not hold: $c^Tx = -4\cdot 11/3 - 2\cdot 4/3 = -52/3$, while $b^T\lambda = 5\cdot -5/3 + 8\cdot -2/3 = -41/3$. They report the correct optimal $x$, but $\lambda$ should be $[-4/3; -4/3]$.
If you replace $-4$ in the objective function with $-3$, the entire example makes sense.
The errata have not been updated for a while. You should contact the authors though, maybe they still update them.
A: I am trying to understand the same example, after I changed the objective function to $-3x_1-2x_2$ I get the same $x_B$ and $\lambda$ as in the book but I get $s_N = [1/3, 5/3]^T$. Can anyone confirm if this is the correct result?
Sorry, I can not add a comment because I don't have enough reputation.
Edit: I found a step-by-step solution for the original objective function $-4x_1-2x_2$ (http://bueler.github.io/M661F16/ex13.1.CORRECTED.pdf)
