Lecture 2 Piecewise-linear optimization

sufficiency: suppose A satisfies the nullspace condition
• let ˆx be k-sparse with support I (i.e., with P Iˆx = ˆx); define y = Aˆx
• consider any feasible x (i.e., satisfying Ax = y), different from ˆx
• define z = x− ˆx; this is a nonzero vector in nullspace(A)
‖x‖ 1 = ‖ˆx+z‖ 1
≥ ‖ˆx+z −P I z‖ 1 −‖P I z‖ 1
= ∑ k∈I
|ˆx k |+ ∑ k∉I
|z k |−‖P I z‖ 1
= ‖ˆx‖ 1 +‖z‖ 1 −2‖P I z‖ 1
> ‖ˆx‖ 1
(line 2 is the triangle inequality; the last line is the nullspace condition)
therefore ˆx = argmin Ax=y ‖x‖ 1
Piecewise-linear optimization 2–18