On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems

Abstract

We present two modified versions of the primal-dual splitting algorithm relying on forward---backward splitting proposed in V $$\tilde{\mathrm{u}}$$ u ~ (Adv Comput Math 38(3):667---681, 2013) for solving monotone inclusion problems. Under strong monotonicity assumptions for some of the operators involved we obtain for the sequences of iterates that approach the solution orders of convergence of $$\mathcal{{O}}(\frac{1}{n})$$ O ( 1 n ) and $$\mathcal{{O}}(\omega ^n)$$ O ( ? n ) , for $$\omega \in (0,1)$$ ? ? ( 0 , 1 ) , respectively. The investigated primal-dual algorithms are fully decomposable, in the sense that the operators are processed individually at each iteration. We also discuss the modified algorithms in the context of convex optimization problems and present numerical experiments in image processing and pattern recognition in cluster analysis.