Homework 4

Deadline: Nov. 2, 22:00:00

First and second order conditions for convexity

  1. The second-order condition tells us that a twice differentiable is strictly convex if has for all , but not vice versa. Here we verify two examples.

    • Show that is a strictly convex function.
    • Show that is a strictly convex function, and there exists such that (i.e., there exists such that ).
  2. For each of the following functions determine whether it is convex, concave, or neither.

    • on .
    • on .
    • on , where .
  3. We've showed that any norm is a convex function. In particular, for all , -norm defined by is a convex function on .

    Now we consider the case where . Is the function convex or concave, or neither? Prove your conclusion.

Convexity-preserving operations

  1. Let on , where is a fixed integer, denotes the vector with , and is the -th largest component of . In other words, are the absolute values of the components of , sorted in nonincreasing order.

    Show that is a convex function.

    (Hint: Express as the pointwise maximum of a family of convex functions.)