A symmetric version of the generalized Chambolle-Pock-He-Yuan method for saddle point problems

A primal-dual method for solving convex-concave saddle point problems was initially proposed by Chambolle and Pock, and its convergence was first proved by He and Yuan later. This primal-dual method (“CPHY" for short) reduces to the Arrow–Hurwicz method (as well as the primal-dual hybrid gradient method) when the combination parameter $$\theta =0$$ θ = 0 , and to a special case of the proximal point algorithm when $$\theta =1$$ θ = 1 , both of which have been well-studied. However, for $$\theta \in (0,1)$$ θ ∈ ( 0 , 1 ) , some theoretical aspects are not yet well-understood, particularly regarding the convergence behavior without imposing strong assumptions. Additionally, although saddle point problems inherently exhibit symmetry between primal and dual variables, the CPHY does not fully exploit this symmetry, as only one variable is updated using an extrapolated step. In this work, we propose a symmetric version of the CPHY by incorporating both symmetry and extrapolation techniques. The resulting algorithm guarantees convergence for $$\theta \in (-1,1)$$ θ ∈ ( - 1 , 1 ) and ensures symmetric updates for both primal and dual variables. Numerical experiments on LASSO, TV image inpainting, and graph cuts demonstrate the algorithm’s improved efficiency.