# Valid Inequality Example (Wolsey Example 9.3)

I was following Wolsey Example 9.3: Let $$X = \{(x,y) \in (R^{m}_+,B^1) : \sum_{i=1}^m x_i \leq my\}$$. Now consider the valid inequality $$x_i \leq y$$ and show that it is facet defining.

My question is why is this a valid inequality? For example one point in $$X$$ may be $$(x,y) = ((0,3,0),(1))$$ such that $$0+3+0 \leq 3\times1$$, but their valid inequality removes this solution, $$3 \leq 1$$.

• Although it is not explicitely written, I believe $x_i \le 1$. Sep 26, 2022 at 10:26
• @Joshua, Are you sure the valid inequality $x_i \leq y$ would not be $x_i \leq M*y$? Sep 26, 2022 at 10:44
• @Omidi That would makes sense to me, but it is not what is written in the book. Sep 26, 2022 at 14:28
• @Kuifje Do you know why it would be implied? Never the less, if I assumed that, the rest of the example would make sense. Sep 26, 2022 at 14:30
• @Joshua I assumed that because with such binaries, $X$ describes part of the facility location problem. Sep 27, 2022 at 8:00

As @Kuifje suggested, an upper bound $$x_i \le 1$$ was mistakenly omitted. This omission was noted in this errata sheet, and it was corrected in the second edition of the book.