Research On Projection Type Algorithms And Inexact Levenberg-Marquardt Type Algorithms For Nonlinear Equations With Applications

Solving the system of nonlinear equations is an important part of the optimization theory and algorithm,and it has been widely applied in many fields of the national economy,such as aerospace,finance management,optimal control,electric power system,to mention just a few.So,designing the efficient method for solving them has always been one of the research hotspots in the fields of optimization and engineering.With the advent of Internet-Big-Data era,the scale of data is growing exponentially.The largescale optimal decision-making problem driven by this has become a scientific problem that needs to be solved urgently in all walks of life.With the help of the rapid improvement of computer performance,the study of efficient numerical algorithms for solving the resulting large-scale and even ultra-large-scale nonlinear equations has important theoretical value and practical significance.This thesis mainly studies the projection type algorithm for solving nonlinear monotone equations(including the unconstrained case and convex constrained case)and the inexact Levenberg-Marquardt(ILM)type algorithm for solving general unconstrained nonlinear equations and their applications.The main contents and innovations for this thesis are summarized as follows.1)Combining the inertial technique,the relaxation technique,the line search technique and the projection technique,an inertial-relaxation projection algorithm framework for solving unconstrained nonlinear monotone equations is proposed.In the algorithm framework,the search direction is only required to satisfy the sufficient descent condition.Under some mild conditions,the global convergence is analyzed.Moreover,two inertia-relaxation projection algorithms satisfying convergence conditions are designed.The numerical experiments about the classical nonlinear monotone equations illustrate the feasibility and effectiveness of these two algorithms.Finally,applying the proposed algorithms to solve the sparse signal restoration problems in compressed sensing verifies that both algorithms are efficient and robust.2)Based on the iterative information of the algorithm,an adaptive line search is designed,and then a derivative-free projection-based algorithm framework for solving convex constrained nonlinear monotone equations is proposed.Under some mild conditions,we establish the global convergence for the just-mentioned algorithm framework.In addition,we show that it has a Q-linear convergence rate under the local error bound condition and the iteration-based error bound condition,respectively.Incorporating two search directions customizing the convergence conditions into the above-mentioned algorithm framework,the numerical experiments on the classical nonlinear monotone equations with convex constraints show the effectiveness and efficiency of the adaptive line search and the resulting algorithms.Further,we apply the presented algorithms to solve the split feasibility problems with wide practical background and the corresponding results illustrate their applicability and robustness.3)An ILM method without restart is presented for solving unconstrained nonlinear equations.When the unit step is not accepted,the proposed method does not need to reset the search direction to the negative gradient direction of the merit function,but directly executes the Armijo-type line search along the former.Under some standard assumptions,we establish the global convergence of the proposed method.Moreover,under the H(?)lderian local error bound condition,we show that our method has superlinear convergence rate,even quadratic convergence rate for some special parameters.The numerical experiments on the underdetermined nonlinear equations and the Tikhonov-regularized logistic regression in statistical learning show that the presented method is efficient and applicable.4)Inspired by the idea of dimensionality reduction,an ILM method with dimensionality reduction is presented.Under some standard assumptions,we show the global convergence of the proposed method.Preliminary numerical results verify the effectiveness of our method.