Assigned Mon Apr 25, due Thurs May 5
Subgradients for Nonconvex Functions
See my notes for the necessary background.
This is not a question, but a comment. The functions given above that are
locally Lipschitz at 0 are the ones for which the horizon subdiffential consists only of 0,
and for these functions, the Clarke subdifferential at 0 is simply the convex hull of the
- 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:
Using the definition on p.6 of the notes, which of these 5 functions is regular at 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)=x2sin(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 C1 at 0.
- Same questions for the following two functions of n variables, at x=0:
Which of these 2 functions is regular at 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)