An $\mathcal O(1/{k})$ Convergence Rate for the Variable Stepsize Bregman Operator Splitting Algorithm

An earlier paper proved the convergence of a variable stepsize Bregman operator splitting algorithm (BOSVS) for minimizing φ(Bu) + H(u), where H and φ are convex functions, and φ is possibly nonsmooth. The algorithm was shown to be relatively efficient when applied to partially parallel magnetic resonance image reconstruction problems. In this paper, the convergence rate of BOSVS is analyzed. When H(u) = ‖Au − f‖2, where A is a matrix, it is shown that for an ergodic approximation uk obtained by averaging k BOSVS iterates, the error in the objective value φ(Buk)+H(uk) is O(1/k). When the optimization problem has a unique solution u∗, we obtain the estimate ‖uk − u∗‖ = O(1/ √ k). The theoretical analysis is compared to observed convergence rates for partially parallel magnetic resonance image reconstruction problems where A is a large dense ill-conditioned matrix.