The Splitting Algorithms And Resolvent Dynamic Systems For Monotone Inclusions

Maximal monotone inclusions are important problems in the fields of optimization ,control,etc. and include as special cases convex minimization,complementary problems and monotone variational inequalities. The operator splitting algorithms, among various available and traditional algorithms for the problems, play an important role, while dy-namic systems are another type of methods for solving the problems in order to satisfy the requirement for real-time solutions. This thesis mainly studies the splitting methods and resolvent dynamic systems for solving maximal monotone inclusions and is divided into three chapters.The first chapter is a brief introduction of maximal monotone inclusions, which in-cludes their state-of-the-art developments. The chapter also provides traditional algorithms and dynamic systems for the problems. In addition, the key points of the thesis is briefly introduced.The second chapter describes the modified operator splitting method for solving a constrained maximal monotone inclusion problem. The request of an operator is weaker than that in the modified algorithm. Its weak convergence is also established. Furthermore, the convergence of the modified method is testified by preliminary numerical results.The third chapter proposes a new resolvent dynamic system for solving an uncon-strained maximal monotone inclusion problem. Its stability in the sense of Lyapunov is proved in theories and by preliminary numerical tests.