 Chambolle-Pock algorithm

For a saddle point problem ${\min }_u{\max }_v\{F(u)+⟨Du,v⟩-G(v)\}$, one update ordering isThe 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 $F(u)+G^{\ast }(Du)$, with $G^{\ast }$ the convex conjugate. A dual-first ordering is obtained by exchanging roles; consistent indexing is needed when comparing the formulas.

The Chambolle-Pock algorithm is a powerful method for solving optimization problems that involve a combination of convex functions and Bregman distances. It is particularly useful for problems that can be framed as finding a minimizer of a convex function subject to certain constraints.