Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 339 / 2 / e / Solution

Write $w_k=(x_k,z_k)$ and $w_{k+1}=(x_{k+1},z_{k+1})$. ExpandinggivesThe off-diagonal terms in the first equation cancel. By the optimality condition for the proximal operator,The second equation then becomes the explicit linear updateThus each step of this preconditioned proximal point algorithm uses only one evaluation of the proximal operator of ${\alpha }^{-1}f$, together with applications of the linear map $A$ and its transpose $A^T$.