Exercise · ZW-PROXIMAL-OPERATOR-004 · ★☆☆ Basic
Which condition characterizes \( p=\operatorname{prox}_{f}(x) \)?
Choose one option.
Show answer
- A. \( x-p\in\partial f(p) \)
Explanation
Optimality of the proximal problem gives \( x-p\in\partial f(p) \). The option with \( \partial f(x) \) evaluates the subdifferential at the wrong point.
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-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-008 For f(u)=|u| and x=-0.5 , 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 ?