Exercise · ZW-SUBGRADIENT-004 · ★★☆ Intermediate
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?
Choose one option.
Show answer
- B. \( (x_1 - x_2)^\top (g_1 - g_2) \ge 0 \)
Explanation
Adding the two subgradient inequalities gives \( (x_1 - x_2)^\top (g_1 - g_2) \ge 0 \), which is the monotonicity of the subdifferential. Reversing the sign is an easy mistake when rearranging.
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-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-008 Let f(x) = |x| + x. What is the subdifferential at x = 0?
- ZW-SUBGRADIENT-009 For f(x) = |x - 3| at x = 3, what subdifferential does f have?