1. Suppose x,r that achieves the minimum has the property that x is not an eigenvector. Can you get a contradiction from this? In particular, can you perturb x to get a smaller r?

Now, the  hardest part of the problem is the last one, namely is to argue that the r that achieves the minimum is the magnitude of the largest eigenvalue in magnitude of A. Here is a trick which I think is useful here. First, show that the largest eigenvalue in magnitude of a positive matrix A with identical row sums s is actually s. Now given an (r,x) that achieve the optimum, can you use them to construct a matrix D such that D^{-1} A D is positive and has identical row sums?

4. You would like to show that two convex sets are identical. A typical strategy is to assume not, apply the separating hyperplane theorem, and obtain a contradiction.

5. The key here just to play around with the definition of a linear variety. You know that quantites like 2x – y or 3x-2y are contained it provided that x,y are contained in it. You can keep taking linear combinations of such things to get still different linear combinations that are contained in it.

Welcome to the website for IE 510! A catalog description of the course may be found here. A tentative syllabus for the course, subject to revision, may be found here. The first homework can be downloaded here; it will be due two weeks after the first lecture on Tuesday, February 3rd, 2015 Thursday, February 5th.Tuesday, Feb 10th.

Update 1/19/2015: I have added a fifth problem to the homework.

Update 1/23/2015: Fixed two issues on the homework. In problem 1, we need to to add the condition that x is a nonnegative, nonzero vector. In problem 3.b., I should have asked you to show the function is concave, not convex. Due to these issues, I have postponed the homework due date by a couple of days.

Update 1/26/2015: I made some minor changes. First, problem 4 needed the condition that the set C is closed. Secondly, I tinkered a little with problem 1 – r should have been e^(r’).

By the way, that there are several variations of the problem – you can maximize r subject to the condition Ax => rx, or minimize r subject to the condition Ax <= rx. Feel free to solve either one for full credit (but if in doubt just go with the one in the pdf of the homework)

Update 1/29/2015: I updated the homework – Problem 4 needs the additional assumption that C contains the origin in its interior.