Multi-step Inertial Algorithm Of Nonlinear Operators And Its Application

The nonlinear problem refers to the problem involving nonlinear operator.Nonlinear problems have received widespread attentions because many problems in scientific research and engineering practice can be transformed into nonlinear problems.The research on the theory and algorithms of nonlinear problems have achieved rich results.In this thesis,we study some algorithms of nonlinear problem by applying the mathematical tools of the bounded perturbation resilience,projection method and other mathematical methods,combining geometry in the Hilbert space,multi-step inertial and fixed point theory in this study.The full text is divided into the following parts:Firstly,we introduce a Krasnosel'skii-Mann(KM)algorithm with perturbations and study the bounded perturbation resilience of KM algorithm.Based on that,we introduce a multi-step inertial KM algorithm(MiKM).We also establish global pointwise and ergodic iteration complexity bounds of MiKM.As an application of MiKM,we construct some multi-step inertial splitting methods to solve the structured monotone inclusion problem.The necessary of introducing multi-step inertial algorithms are supported by the numerical simulation.Secondly,we study a projection method with outer perturbations for the split equality problem(SEP)and establish its weak convergence.Then we introduce a relaxation projection algorithm and two approximate algorithms to solve the SEP,since the projection algorithm involves two projections on a closed convex set,which is generally difficult.The effectiveness of the algorithms is tested by the numerical simulation.Finally,we study two types of special equilibrium problems.First,nash equilibrium models.We introduce two projection algorithms for solving the models by combining the projection method for equilibrium problem and the gradient method for the inclusion ? .Second,split feasibility problem involving paramonotone equilibria and convex optimization problem.We introduce a new algorithm which involves projecting each iteration to solve a split feasibility problem with paramonotone equilibria and using unconstrained convex optimization.Then,we replace the unconstrained convex optimization by a constrained convex optimization and introduce some algorithms for two different types of objective function of the constrained convex optimization.The convergence of the algorithms is proved.And by some numerical experiments,we show that these algorithms are very effective in solving practical problems.