Lecture 18 Subgradients

Lecture 18
• The function f(x) = max{x 2 + 2x − 3, x 2 − 2x − 3, 4}
f(x) =
⎧
⎪⎨
⎪⎩
x 2 − 2x − 3
for x < −1
4 for x ∈ [−1, 1]
x 2 + 2x − 3 for x > 1
∂f(x) =
⎧
⎪⎨
⎪⎩
2x − 2 for x > 1
[−4, 0]
for x = −1
0 for x ∈ (−1, 1)
[0, 4] for x = 1
2x + 2 for x > 1
• The optimal set is X ∗ = [−1, 1]
• For every x ∗ ∈ X ∗ , we have 0 ∈ ∂f(x ∗ )
Convex Optimization 17