Solving Composite Fixed Point Problems with Block Updates

Abstract

Various strategies are available to construct iteratively a common fixed point of nonexpansive operators by activating only a block of operators at each iteration. In the more challenging class of composite fixed point problems involving operators that do not share common fixed points, current methods require the activation of all the operators at each iteration, and the question of maintaining convergence while updating only blocks of operators is open. We propose a method that achieves this goal and analyze its asymptotic behavior. Weak, strong, and linear convergence results are established by exploiting a connection with the theory of concentrating arrays. Applications to several nonlinear and nonsmooth analysis problems are presented, ranging from monotone inclusions and inconsistent feasibility problems, to minimization problems arising in data science.

Nonlinear Forward-Backward Splitting with Momentum Correction

“…Two examples of existing algorithms that can be interpreted as instances of Algorithm 1 are the forward-half-reflected-backward (FHRB) method and its special case, the forward-reflected-backward (FRB) method 2 [22]. FHRB is a method for finding x ∈ H such that 0 ∈ Bx + Dx + Cx (9) for which the following assumption holds; FRB solves the same problem but with C = 0.…”

“…In this special case, we do not need to evaluate both M k−1 x k and M k x k fully since we can reuse the computation of Dx k . Algorithm 3 Forward-Half-Reflected-Backward [22] Consider problem (9). With x 0 , x −1 ∈ H and α −1 > 0, for all k ∈ N iteratively perform…”

“…Corollary 5.1. Let Assumption 5.1 hold and consider problem (9) and Algorithm 3. If there exists ǫ > 0 such that…”

“…Consider again problem (9) and the operator choices that generated FHRB; (9). With x 0 , x −1 ∈ H and α −1 > 0, for all k ∈ N iteratively perform…”

“…If the resolvent (Id +A) −1 of A is easily computable, this problem can be solved with the forward-backward splitting method [1,2]. Since this might not be the case, great effort has been devoted to constructing other splitting methods that can exploit an additional structure of A, sometimes further assuming C = 0 [3][4][5][6][7][8][9]. This work presents an alternative approach for analyzing and constructing such splitting methods by formulating them as different instances of a forward-backward method with a nonlinear resolvent (M + A) −1 • M where M : H → H is a (potentially) nonlinear kernel.…”