Lecture 2 Piecewise-linear optimization

l 1 -Norm approximation
minimize ‖Ax−b‖ 1
• l 1 -norm of m-vector y is
‖y‖ 1 =
m∑
|y i | =
i=1
m∑
max{y i ,−y i }
i=1
• equivalent LP (with variable x and auxiliary vector variable u)
minimize
m∑
u i
i=1
subject to −u ≤ Ax−b ≤ u
(for fixed x, optimal u is u i = |(Ax−b) i |, i = 1,...,m)
Piecewise-linear optimization 2–9