Inertial Splitting Methods For Monotone Inclusions Of Three Operators

Convex minimization problem,monotone complementary problem and monotone variational inequality problem are fundamental problems in many fields,such as operation research,management,optimization and control.All these problems can be reduced to monotone inclusions.For methods of solving such problems,the operator splitting methods show powerful computational ability.Among them,the forward-backward splitting method,Peaceman/Douglas-Rachford splitting method and Tseng splitting method and so on are the most popular methods.This thesis mainly discusses inertial splitting methods for monotone inclusions of three operators in infinite-dimensional real Hilbert spaces,the main contents are described as follows:In the first chapter,we briefly introduce the research status of monotone inclusions,as well as the main points of this thesis,and give the relevant preliminary knowledge.In the second chapter,we focus on the inertial Douglas-Rachford operator splitting method for monotone inclusions of three operators,and by introducing the characteristic operator via Attouch-Thera duality principle,we develop more self-contained and less convoluted techniques to prove its weak convergence under mild assumptions.In the third chapter,we mainly consider the linearly composed convex minimization problem.We design recursive formulae by optimality condition,and prove weak convergence of the resulting inertial splitting method.In the fourth chapter,we do numerical experiments to verify that introducing inertial term can indeed improve numerical results in some cases.