= Chambolle–Pock algorithm
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{wiki}
= Primal-dual hybrid gradient method
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= PDHG
{c}
{synonym}
For a <saddle point> problem $\min_u\max_v\{F(u)+\langle Du,v\rangle-G(v)\}$, one update ordering is
$$
u^{k+1}=\operatorname{prox}_{\tau F}(u^k-\tau D^*v^k),\qquad v^{k+1}=\operatorname{prox}_{\sigma G}(v^k+\sigma D(2u^{k+1}-u^k)).
$$
The <proximal operators> treat both nonsmooth convex terms, while the extrapolation $2u^{k+1}-u^k$ couples the primal and dual variables. The primal objective is $F(u)+G^*(Du)$, with $G^*$ the <convex conjugate>. A dual-first ordering is obtained by exchanging roles; consistent indexing is needed when comparing the formulas.
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