Comments on Midterm - Winter 2005 (not complete yet)

The first part of the course has been on convex analysis and has been more on the theoretical side. We will now (post-midterm) start to learn more about algorithms and applied problems.

Comments on Midterm now Follow

1. 1. Krein-Milman: See Course Notes.
   2. The form of the polyhedral set P means that P is a finite intersection of closed halfspaces (use Ax &le b and continuity). Therefore, P is closed and bounded and so compact, and is convex. Now apply Krein-Milman as on the assignment.
2. 1. One must use the fact that the first function g is convex and not just monotonic!
   2. One must show e^x is convex.
   3. Give a counterexample to the case when Q is not positive definite, e.g. Q=0 works.
3. 1. x=0 is NOT the only critical point; but x=0 is a strict local minimum.
      Please review gradient and Hessian - many were not clear on how to find these.
      Because the midterm stated that there was a unique critical point, please bring your paper to me for remarking.
4. 1. Review the differences between necessary and sufficient optimality conditions. This is vital for the course. (It will definitely 'reappear' on the final exam.)
5. 1. This problem was done in class (but with the translation x₁ becomes x₁-1). So optimum is at x₁=1 and x₂=0.