Lecture 2 Piecewise-linear optimization

Proof of nullspace condition
notation
• x has support I ⊆ {1,2,...,n} if x i = 0 for i ∉ I
• |I| is number of elements in I
• P I is projection matrix on n-vectors with support I: P I is diagonal with
(P I ) jj =
{
1 j ∈ I
0 otherwise
• A satisfies the nullspace condition if
‖P I z‖ 1 < 1 2 ‖z‖ 1
for all nonzero z in nullspace(A) and for all support sets I with |I| ≤ k
Piecewise-linear optimization 2–17