NONLINEAR FORWARD-BACKWARD SPLITTING WITH PROJECTION CORRECTION

被引:16
|
作者
Giselsson, Pontus [1 ]
机构
[1] Lund Univ, Dept Automat Control, SE-22100 Lund, Sweden
基金
瑞典研究理事会;
关键词
Key words; monotone inclusions; nonlinear resolvent; forward-backward splitting; forward-backward-forward splitting; four-operator splitting; PROXIMAL POINT ALGORITHM; MONOTONE-OPERATORS; 1ST-ORDER METHODS; CONVERGENCE; OPTIMIZATION; SUM; EXTRAGRADIENT; INCLUSIONS;
D O I
10.1137/20M1345062
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We propose and analyze a versatile and general algorithm called nonlinear forwardbackward splitting (NOFOB). The algorithm consists of two steps; first an evaluation of a nonlinear forward-backward map followed by a relaxed projection onto the separating hyperplane it constructs. The key to the method is the nonlinearity in the forward-backward step, where the backward part is based on a nonlinear resolvent construction that allows for the kernel in the resolvent to be a nonlinear single-valued maximal monotone operator. This generalizes the standard resolvent as well as the Bregman resolvent, whose resolvent kernels are gradients of convex functions. This construction opens up a new understanding of many existing operator splitting methods and paves the way for devising new algorithms. In particular, we present a four-operator splitting method as a special case of NOFOB that relies on nonlinearity and nonsymmetry in the forward-backward kernel. We show that forward-backward-forward splitting (FBF), forward-backward-half-forward splitting (FBHF), and asymmetric forward-backward-adjoint splitting with its many special cases are special cases of the four-operator splitting method and hence of NOFOB. We also show that standard formulations of FB(H)F use smaller relaxations in the projections than allowed in NOFOB. Besides proving convergence for NOFOB, we show linear convergence under a metric subregularity assumption, which in a unified manner shows (in some cases new) linear convergence results for its special cases.
引用
收藏
页码:2199 / 2226
页数:28
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