Lecture 2 Piecewise-linear optimization

necessity: suppose A does not satisfy the nullspace condition
• for some nonzero z ∈ nullspace(A) and support set I with |I| ≤ k,
‖P I z‖ 1 ≥ 1 2 ‖z‖ 1
• define a k-sparse vector ˆx = −P I z and y = Aˆx
• the vector x = ˆx+z satisfies Ax = y and has l 1 -norm
‖x‖ 1 = ‖−P I z +z‖ 1
= ‖z‖ 1 −‖P I z‖ 1
≤ 2‖P I z‖ 1 −‖P I z‖ 1
= ‖ˆx‖ 1
therefore ˆx is not the unique l 1 -minimizer
Piecewise-linear optimization 2–19