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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.

Rodrigo de Azevedo
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jjjjjj
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    There's also a typo in the second iteration (with basis set = ${3, 1}$): "$d=(3/2, -1/2)$" is wrong; it should be $d=(3/4, 1/4)$, otherwise $x_2^+ = 4/3$ won't make sense. – Yibo Yang Apr 03 '19 at 21:07

2 Answers2

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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.

LinAlg
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    For the example to make sense -- wanted to confirm that I think you mean: replace $-4x_1 + \ldots$ with $-3x_1 + \ldots$ in the objective function, right? When I work through the problem modified in this way, I get things lining up with NW's answer. For duality, if you do $-3\cdot 11/3 - 2\cdot 4/3$, then it should work out ok – jjjjjj Mar 06 '18 at 04:32
  • You are right, I edited my answer. – LinAlg Mar 06 '18 at 14:51
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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)

Ganymed_
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