Consider the following claim:

  • Suppose you have two distinct valid inequalities \(ax \le b\) and \(cx \le d\). Further suppose that your polyhedron is full-dimensional. Then the aggregated inequality \( (a+c)x \le (b+d) \) cannot induce a facet.

This claim is true, and I have used it in my research. However, I was hoping to find it explicitly stated in literature. Other researchers have referred me to Schrijver's Theory of LP and IP, but I cannot seem to find it. (Perhaps it is too trivial a result to include.) Is this claim made in any published material?

asked 29 Jul '14, 16:30

Austin%20Buchanan's gravatar image

Austin Buchanan
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accept rate: 42%


It seems to me that this is an easy consequence of the fact that the set of facets forms a minimal description of the polyhedron, so removing one must produce a strictly larger polyhedron. That can't be true of an aggregated inequality if the component inequalities are part of the description.

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answered 30 Jul '14, 19:25

Matthew%20Saltzman's gravatar image

Matthew Salt... ♦
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accept rate: 17%

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Asked: 29 Jul '14, 16:30

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Last updated: 30 Jul '14, 19:25

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