Samstag, Februar 28, 2015

Interesting mistakes

I taught a course in Linear and Integer Optimization last semester, the exam of which was held last week. I'm fairly pleased with the results. Grading an exam is often painful when you realize that some students failed to understand important concepts, but overall it appears that my expectations about the difficulty of this exam were mostly accurate. Sometimes, however, students fail at a problem for reasons that we do not anticipate. One problem of the exam was to prove that when P and Q are (convex) polytopes, then so is their Minkowski sum P + Q.

I am aware of two reasonable approaches to this problem. The first is to use inequality representations to argue that the product P x Q is a polyhedron, and then appeal to the fact (shown in the lecture) that the image under the linear map (p, q) -> p + q is a polyhedron (boundedness is of course easy to see, and naturally there are some variations of this approach). The second approach is to use representations of P and Q as convex hulls of finitely many points, and to show that P + Q is the convex hull of the pairwise sum of such points. This second proof is elementary but requires a bit more work because one has to argue about the coefficients in convex combinations.

To my surprise, several students attempted a third route. They started with inequality representations and used the boundedness to argue that given one of the inequalities ax <= b that define P, the maximum of ax over Q is finite. They then tried to prove that P + Q can be obtained by simply using both the inequalities defining P and those defining Q together, with the right-hand sides increased by the corresponding (finite) maximum over the other polytope. This approach cannot possibly work because the number of facets of the Minkowski sum can be exponentially larger than the number of facets of the summands (the permutahedron is a cute example of this). And yet, in hindsight, it is a very plausible approach to try if one doesn't know about this fact.

As mistakes go, I would call this an interesting mistake. In some sense it isn't even a mistake at all, just a false trail into the woods that unfortunately some students got lost in.

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