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I read this theorem in a book about Linear Optimization:

In the simplex method, for a minimization problem, a BFS is optimal if all of the reduced costs are negative, i.e. $\forall i \quad z_i-c_i \le 0$.

I am curious is the inverse of the above theorem also true? Or can we find an optimal solution such that $z_k-c_k>0$ for some $k$? The latter case means the simplex algorithm does not necessarily terminate when it reaches an optimal solution.

Any help is appreciated.

Masoud
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  • Could you be more specific by what you mean by inverse? Do you mean converse? –  Oct 03 '22 at 01:07
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    @PseudoLooped I do not know what you mean by the "Converse". I am asking if is there a BFS with some positive reduced cost. – Masoud Oct 03 '22 at 16:48

1 Answers1

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The only situation this would be true if there exists a constraint that causes degeneracy in a model where one of the basic variables has a zero as its right-hand-side value, and thus doesn’t contribute to a model upon a pivot. For example:

Suppose we have a model:

$$\text{min }z=-x_1+x_2-x_3$$

Subject to,

$$x_1+x_2\le 4$$ $$-x_2+x_3\le0$$ $$x_1,x_2,x_3 \ge0$$

Converting this to standard form, we get:

$$\text{min } z +x_1 - x_2 +x_3 = 0$$

Subject to: $$x_1+x_2+s_1=4$$ $$-x_2+x_3+s_2=0$$ $$x_1, x_2, x_3, s_1, s_2 \ge0$$

From here, lets put this in a tableau:

enter image description here

Let’s pivot the $x_1$ column to produce:

enter image description here

Then lets pivot the $x_3$ column to produce our final tableau: enter image description here

Notice that the solution produced in the second tableau is optimal, $(4,0,0)$, and is exactly the same as the solution produced by the third tableau. In addition, the second tableau produced an optimal solution, but had a $C^\pi_j > 0$, which shows that the simplex method doesn’t terminate right away.

Here’s a PowerPoint slide I found that explains more on Degeneracy in models.

JJMae
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