Lecture 2 Piecewise-linear optimization

l ∞ -Norm (Cheybshev) approximation
with A ∈ R m×n , b ∈ R m
minimize ‖Ax−b‖ ∞
• l ∞ -norm (Chebyshev norm) of m-vector y is
‖y‖ ∞ = max
i=1,...,m |y i| = max
i=1,...,m max{y i,−y i }
• equivalent LP (with variables x and auxiliary scalar variable t)
minimize t
subject to −t1 ≤ Ax−b ≤ t1
(for fixed x, optimal t is t = ‖Ax−b‖ ∞ )
Piecewise-linear optimization 2–7