GRPDA Revisited: Relaxed Condition and Connection to Chambolle-Pock’s Primal-Dual Algorithm

Abstract

Recently, a golden ratio primal-dual algorithm (GRPDA) was proposed by Chang and Yang for solving structured convex optimization problems. It is a new adaptation of the classical Arrow-Hurwicz method by using a convex combination step, instead of the widely adopted extrapolation technique. The convex combination step is determined by a parameter $\psi$, which, to guarantee global convergence, is restricted to $(1, (1+\sqrt{5})/2]$. In this paper, by carrying out a refined analysis, we expand this region to $(1, 1+\sqrt{3})$. Moreover, we establish ergodic sublinear convergence rate results based on function value residual and constraint violation of an equivalently reformulated constrained optimization problem, rather than the previously adopted so-called primal-dual gap function that could vanish at nonstationary points. For linear equality constrained and regularized least-squares problems, we further show that GRPDA and Chambolle-Pock’s primal-dual algorithm are equivalent provided that some parameters are chosen properly. Finally, experimental results on the LASSO and the basis pursuit problems are presented to demonstrate the performance of GRPDA with $\psi$ being chosen in the expanded region.