Due: Friday, September 23, 2011, in class.

Let C = {x : x₁² + x₂² + x₃² ≤ 1} and y = (1, 0, 2)^T. Find the minimum distance from y to C, the unique minimizing point, and a separating hyperplane.
Let $x + x ≤ 1, 3 1 2 S = {x ∈ IR : x1 + x3 ≤ 1, xi = 0 or 1, i = 1,...,3 } x2 + x3 ≤ 1,$
Show that the point = 0.5(1, 1, 1)^T is not in the convex hull of S, by finding a hyperplane which separates the point from the set.
Let C be the open interval (-α,α). Let f(x) := (α² - x²)^-. Show that f is convex on C.
Let aⁱ, i = 1,…,m be vectors in IRⁿ and let b_i, i = 1,…,m be scalars. Let α be a positive scalar. Define the function f : IR → IR as ${ f(z) := |z| - α if |z| > α 0 if |z| ≤ α$
Show that the function g(x) := ∑ _i=1^mf(b_i - aⁱ^Tx) is convex.
Consider the problem min{f(x) := (x - 1)⁴ + e^x}.
1. Show that f(x) is a strictly convex function.
2. We want to get a lower bound on the optimal value of this problem. Consider the subgradient inequality $T f (x ) ≥ f(¯x) + ∇f (¯x) (x - ¯x ).$
  Apply this inequality at = -1, 0, and 1, and construct a linear program to find a lower bound on the optimal value of the problem.
Let g : IR₊ → IR be a strictly increasing nonnegative function on the set of positive real numbers. Is xg(x) a convex function of x > 0? Prove or give a counterexample.

John Mitchell

Amos Eaton 325

x6915.

Office hours: Tuesday 12.0 – 2.0, Wednesday 2.0 – 4.0.