Exercise · ZW-PROXIMAL-GRADIENT-METHOD-010 · ★★★ Advanced
Minimize \( \tfrac12(x-6)^2+|x| \) over \( \mathbb{R} \) with one proximal gradient step using \( t=1 \) from \( x_0=0 \). What is \( x_1 \)?
Choose one option.
Show answer
- B. \( 5 \)
Explanation
The gradient at \( 0 \) is \( -6 \), so \( z=6 \). Applying soft thresholding with \( t\lambda=1 \) gives \( 5 \). Not applying the threshold gives \( 6 \).
Review the concept: Proximal gradient method
Proximal gradient minimizes \( f(x)+g(x) \), with \( f \) convex and smooth and \( g \) convex but possibly nondifferentiable, alternating a gradient step on \( f \) and a proximal step on \( g \).
\[x_{k+1}=\operatorname{prox}_{t_k g}\left(x_k-t_k\nabla f(x_k)\right)\]
More practice problems: proximal gradient method
- ZW-PROXIMAL-GRADIENT-METHOD-001 Which function g appears in the proximal gradient method for minimizing f(x)+g(x) when f is smooth?
- ZW-PROXIMAL-GRADIENT-METHOD-002 Which operation is applied to g in each iteration of the method?
- ZW-PROXIMAL-GRADIENT-METHOD-003 If g=0 , which method does proximal gradient reduce to?
- ZW-PROXIMAL-GRADIENT-METHOD-004 What condition must the gradient of f satisfy for standard convergence guarantees?
- ZW-PROXIMAL-GRADIENT-METHOD-005 For g(x)=λ\|x\|_1 , what does the proximal operator produce componentwise?
- ZW-PROXIMAL-GRADIENT-METHOD-006 If f(x)=tfrac12\|x-a\|^2 and g=0 , what is the result of one step with t=1 from x_k ?
- ZW-PROXIMAL-GRADIENT-METHOD-007 If g(x)=λ|x| in one dimension and the point z=0.3 with tλ=0.5 , what is prox_tg(z) ?
- ZW-PROXIMAL-GRADIENT-METHOD-008 With f(x)=tfrac12 x^2 and g(x)=|x| in one dimension, what is the iterate after one step with t=1 fr…