Exercise · ZW-SUBGRADIENT-008 · ★★☆ Intermediate
Let f(x) = |x| + x. What is the subdifferential at x = 0?
Choose one option.
Show answer
- B. The interval [0, 2]
Explanation
Since f = 2max(x, 0), the subdifferential at 0 is 2 times [0, 1], that is [0, 2]. Forgetting the factor 2 when adding slopes gives the interval [-1, 1].
Review the concept: Subgradient
The subgradient generalizes the gradient to convex functions that are not differentiable: at each point it is a vector g that gives an affine lower bound of f, that is, f(y) ≥ f(x) + gᵀ(y − x) for all y.
\[f(y) \geq f(x) + g^\top (y - x)\]
More practice problems: subgradient
- ZW-SUBGRADIENT-001 What is the name of the set of all subgradients of f at x?
- ZW-SUBGRADIENT-002 If f is differentiable at x, what is the subdifferential of f at that point?
- ZW-SUBGRADIENT-003 For f(x) = |x|, what is the subdifferential at x = 0?
- ZW-SUBGRADIENT-004 If g_1 is a subgradient of f at x_1 and g_2 is one at x_2 , with f convex, which inequality holds?
- ZW-SUBGRADIENT-005 What does it mean for g to be a subgradient of f at x?
- ZW-SUBGRADIENT-006 For f(x) = max(x, 0), what is the subdifferential at x = 0?
- ZW-SUBGRADIENT-007 For f(x) = 2|x| at x = 1, what is the subdifferential?
- ZW-SUBGRADIENT-009 For f(x) = |x - 3| at x = 3, what subdifferential does f have?