Assigned Mon Apr 25, due Thurs May 5

See my notes for the necessary background.

- We say that x locally minimizes f if, for all sufficiently small z, f(x+z) ≥ f(x). Show that it follows that a necessary condition for x to locally minimize f is that 0 is in the regular subdifferential of f at x, immediately from the definition on the first page of the notes.
- Determine the regular subdifferential, the (general) subdifferential and the horizon subdifferential
of the following functions of one variable at x=0:
- f(x)=|x|
^{ 3} - f(x)=|x|
^{ 1/3} - f(x)=a|x| where a is nonnegative real number
- f(x)=a|x| where a is a negative real number
- f(x)=x
^{2}sin(1/x) if x is nonzero, f(0)=0. You can check by using the definition of the derivative that f is differentiable at 0 with f′(0)=0, and you can get f′(x) at any other x by using the ordinary rules of calculus. Verify that f′ is not continuous at 0, and hence that f is not C^{1}at 0.

- f(x)=|x|
- Same questions for the following two functions of n variables, at x=0:
- f(x) = 3rd largest entry of x, assuming n ≥ 3 (see p. 3-5 of the notes)
- f(x) = largest entry of Ax, where A is any n by n matrix (use the chain rule on p.7 of the notes)