Let f(u)=λ|u| with λ=1 . Compute prox_f(x) for x=-2.5 and decide which statement is true when checking x-p∈∂ f(p) .

p=-1.5 and x-p=-1 ∈∂ f(-1.5) , so it holds

p=-1.5 and x-p=-1 ∈∂ f(-1.5) is false

p=-1.5 and x-p=1 ∈∂ f(-1.5) , so it holds

Exercise · ZW-PROXIMAL-OPERATOR-010 · ★★★ Advanced

Let \( f(u)=\lambda|u| \) with \( \lambda=1 \). Compute \( \operatorname{prox}_{f}(x) \) for \( x=-2.5 \) and decide which statement is true when checking \( x-p\in\partial f(p) \).

Choose one option.

Show answer

C. \( p=-1.5 \) and \( x-p=-1 \in\partial f(-1.5) \), so it holds

Explanation

Thresholding gives \( p=-1.5 \); since \( p<0 \), \( \partial f(p)=\{-1\} \), which contains \( x-p=-1 \). The option with \( x-p=1 \) reverses the sign of the subdifferential.

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Review the concept: Proximal operator

The proximal operator of a convex function \( f \) maps each point \( x \) to the unique minimizer of \( f(u)+\frac{1}{2}\|u-x\|^2 \).

\[\operatorname{prox}_{f}(x)=\arg\min_{u}\left\{f(u)+\tfrac{1}{2}\|u-x\|^2\right\}\]

Let \( f(u)=\lambda|u| \) with \( \lambda=1 \). Compute \( \operatorname{prox}_{f}(x) \) for \( x=-2.5 \) and decide which statement is true when checking \( x-p\in\partial f(p) \).

Explanation

Review the concept: Proximal operator

More practice problems: proximal operator