Due: Mon, Sept 18th in class.

You can work on this problem set with other students. If you do, write down the names of all who you talked to.

- A function is convex if for any and any , the function satisfies Show that any local minimum of such a function (smallest in some neighborhood) is also a global minimum.
- Give an example where the Newton iteration does not converge.
- Show that the function for a convex set is a convex function.
- Give an example where projected gradient descent applied to a convex function and a convex set takes iterations to reach a point whose function value is within of the minimum.
- (bonus) Suggest an idea to speed up gradient descent and justify your answer.