求解单调包含的四算子分裂算法

Four-operator Splitting Algorithms for Solving Monotone Inclusions

近年来,含有三个及以上的极大单调算子之和的单调包含问题备受关注.本文提出三种求解四个单调算子之和的零点问题的分裂算法,这四个算子分别是两个极大单调算子、一个单调Lipschitz算子和一个余强制算子.所提分裂算法主要基于向前—反射—Douglas-Rachford分裂算法、向后—向前—反射—向后分裂算法和向后—反射—向前—向后分裂算法提出.作为应用,将所提出算法应用于求解极大单调算子有限和的单调包含问题.在凸集的Minkowski和上投影的数值结果验证了所提出算法的有效性.

Monotone inclusions involving the sum of three maximally monotone operators or more have received much attention in recent years.In this paper,we propose three splitting algorithms for finding a zero of the sum of four monotone operators,which are two maximally monotone operators,one monotone Lipschitz operator,and one cocoercive operator.These three splitting algorithms are based on the forward-reflected-Douglas-Rachford splitting algorithm,backward-forward-reflected-backward splitting algorithm,and backward-reflectedforward-backward splitting algorithm,respectively.As applications,we apply the proposed algorithms to solve the monotone inclusions problem involving a finite sum of maximally monotone operators.Numerical results on the projection on Minkowski sums of convex sets demonstrate the effectiveness of the proposed algorithms.