Convergence Rates for Projective Splitting

Abstract

Projective splitting is a family of methods for solving inclusions involving sums of maximal monotone operators. First introduced by Eckstein and Svaiter in 2008, these methods have enjoyed significant innovation in recent years, becoming one of the most flexible operator splitting frameworks available. While weak convergence of the iterates to a solution has been established, there have been few attempts to study convergence rates of projective splitting. The purpose of this paper is to do so under various assumptions. To this end, there are three main contributions. First, in the context of convex optimization, we establish an O(1/k) ergodic function convergence rate. Second, for strongly monotone inclusions, strong convergence is established as well as an ergodic O(1/ √ k) convergence rate for the distance of the iterates to the solution. Finally, for inclusions featuring strong monotonicity and cocoercivity, linear convergence is established.

Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems

“…With our assumption in (33), the fact c 1 = τ µ + ε, and the fact c τ +1 = ε > 0, (39) shows that ξ k+1 ≤ ξ k , for all k ∈ N, from which the result follows.…”

“…The diversity of the Krasnosel'skiȋ-Mann iteration, to which our work is related as mentioned in the sequel, makes it an interesting and valuable tool, see, e.g., [28] in this journal. Research activity on asynchronous algorithms is receiving ever growing attention recently as evidenced by some very interesting recent works of Combettes, Eckstein and coworkers [26,27,29,39,40].…”

Projective splitting with forward steps only requires continuity

“…The stepsize assumptions differ from [14,13] for i ∈ I F in that we no longer assume Lipschitz continuity nor that the stepsizes are bounded by the inverse of the Lipschitz constant. However, the initial trial stepsize for the backtracking linesearch at each iteration is assumed to be bounded from above and below:…”

“…Substituting this expression into (13) and applying the Cauchy-Schwarz inequality to the inner product yields that the left-hand size of (12) is lower bounded by…”

“…Our recent work in [14] incorporated forward steps into the projective splitting framework, in place of the customary proximal (backward) steps, for any Lipschitz continuous operators, and introduced backtracking and adaptive stepsize rules; see also [21]. The even more recent work [13] derived convergence rates for the method under various conditions.…”