Unified linear convergence of first-order primal-dual algorithms for saddle point problems

Document Type

Article

Publication Date

7-1-2022

Abstract

In this paper, we study the linear convergence of several well-known first-order primal-dual methods for solving a class of convex-concave saddle point problems. We first unify the convergence analysis of these methods and prove the O(1/N) convergence rates of the primal-dual gap generated by these methods in the ergodic sense, where N counts the number of iterations. Under a mild calmness condition, we further establish the global Q-linear convergence rate of the distances between the iterates generated by these methods and the solution set, and show the R-linear rate of the iterates in the nonergodic sense. Moreover, we demonstrate that the matrix games, fused lasso and constrained TV-ℓ2 image restoration models as application examples satisfy this calmness condition. Numerical experiments on fused lasso demonstrate the linear rates for these methods.

Publication Source (Journal or Book title)

Optimization Letters

First Page

1675

Last Page

1700

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