Exercise · ZW-PROXIMAL-OPERATOR-008 · ★★☆ Intermediate
For \( f(u)=|u| \) and \( x=-0.5 \), what is \( \operatorname{prox}_{f}(x) \)?
Choose one option.
Show answer
- A. \( 0 \)
Explanation
Since \( |x|\le 1 \), soft-thresholding gives \( 0 \). The option \( -0.5 \) corresponds to not applying the penalty.
Review the concept: Proximal operator
The proximal operator of a convex function \( f \) maps each point \( x \) to the unique minimizer of \( f(u)+\frac{1}{2}\|u-x\|^2 \).
\[\operatorname{prox}_{f}(x)=\arg\min_{u}\left\{f(u)+\tfrac{1}{2}\|u-x\|^2\right\}\]
More practice problems: proximal operator
- ZW-PROXIMAL-OPERATOR-001 What is the value of prox_f(x) for f(u)=0 ?
- ZW-PROXIMAL-OPERATOR-002 For f(u)=(1)/(2)u^2 , what is the proximal operator at x ?
- ZW-PROXIMAL-OPERATOR-003 What is the proximal operator at x=0.5 when f(u)=|u| ?
- ZW-PROXIMAL-OPERATOR-004 Which condition characterizes p=prox_f(x) ?
- ZW-PROXIMAL-OPERATOR-005 What is prox_f(x) if f(u)=2u and x=1 ?
- ZW-PROXIMAL-OPERATOR-006 For f(u)=λ|u| with λ=2 , what is prox_f(-5) ?
- ZW-PROXIMAL-OPERATOR-007 If f(u)=(1)/(2)u^2 and x=4 , what is prox_f(x) ?
- ZW-PROXIMAL-OPERATOR-009 What is prox_f(x) for f(u)=λ|u| with λ=1 and x=0.8 ?