Iterative Solution Of Several Absolute Value Equations And Nonlinear Complementarity Problems

Many practical problems in scientific computing and engineering applications,such as production management,transportation,financial engineering,industrial and agricultural production,can be reduced to solving complex systems involving absolute value functions.Most of these problems are complex optimization prob-lems that are non-differentiable,multi-variable and multi-parameter.Due to their complex structural characteristics,effective numerical solutions for these systems have been difficult to obtain.With the rapid development of social economy and science and technology,the increasing dimensionality of variables in such prob-lems has made their complexity even more challenging to solve,making it crucial to design efficient and robust numerical methods based on the system’s structural characteristics.In a word,how to design an economical,efficient and robust nu-merical solution according to the structural characteristics of the system itself is of great practical significance.This dissertation focuses on the numerical solution of three important systems involving absolute value functions,namely absolute value equations,nonlinear complementarity problems,and tensor absolute value equations.Developing fast and efficient numerical methods for solving these problems is of great theoretical and practical importance.The dissertation consists of three parts.The first part investigates the effective iterative methods for solving absolute value equations.Firstly,an effective iterative method called modified general-ized SOR（MGSOR）is proposed for solving general absolute value equations by transforming them into structured 2×2 block linear systems.The convergence properties of the MGSOR method are analyzed,and the explicit expression of op-timal iterative parameters is given.The MGSOR method inherits the advantages of the generalized SOR（GSOR）method for solving 2×2 block linear systems and the SOR method for solving general absolute value equations,and improves their convergence properties.When the MGSOR method is used to solve gener-al absolute value equations,its numerical performance is stable and superior to some existing effective algorithms.Secondly,a matrix splitting-based momentum accelerated iterative method is designed for solving generalized absolute value e-quations,which combines the splitting technique with momentum acceleration to improve performance.The method is flexible and applicable to various cases,and its convergence properties are analyzed when the coefficient matrix is positive def-inite or₊-matrix,and its optimal momentum factor is determined.Numerical experiments demonstrate the effectiveness of the proposed methods in comparison with existing algorithms such as Picard method,Newton-like method and Newton-based matrix splitting（NMS）for solving generalized absolute value equations.The second part focuses on developing effective iterative methods for solving a class of nonlinear complementarity problems（NCP（F））.The（NCP（F））is first transformed into an equivalent implicit fixed-point equation system,and then a generalized MMS（GMMS）iterative method for（NCP（F））is proposed by intro-ducing two parameter matricesΩ₁andΩ₂and combining with the modulus-based matrix splitting（MMS）iterative method for solving this type of problems.A more efficient preconditioned GMMS（PGMMS）iterative method than GMMS is con-structed by using the idea of preconditioned MMS（PMMS）iterative method for solving linear complementarity problems and selecting an appropriate precondi-tioning matrix.Finally,a preconditioned generalized two-step modulus-based accelerated overrelaxation（PGTMAOR）iterative method is developed by com-bining PGMMS with a two-step MMS（TMMS）iterative method for further im-proving computational efficiency.This method not only combines the skills of preconditioning and two-step iterative methods to improve algorithm efficiency but also provides a universal form based on the MMS iterative method for solving this type of（NCP（F））.The convergence properties of the algorithm are analyzed in detail when the coefficient matrix is positive definite or₊-matrix,and nu-merical experiments are used to compare PGTMAOR with some existing iterative methods,demonstrating the feasibility and effectiveness of the proposed method.The third part investigates the effective iterative methods for solving a class of tensor absolute value equations（TAVE）.Two effective iterative methods are pro-posed based on the specific structural characteristics of TAVE.The first method is derived from the commonly used Anderson acceleration（AA）technology.By combining the tensor splitting（TS）iterative method of multilinear systems with the preconditioned technique,a preconditioned tensor splitting（PTS）iterative method is constructed for solving TAVE.To further improve computational effi-ciency,an Anderson acceleration-based preconditioned tensor splitting（AAPTS）iterative method is designed.The convergence properties of TS and PTS itera-tive methods are analyzed based on the principle of compressed mappings.Based on the theory of the strong semi-smoothness of absolute value functions,the lo-cal convergence properties of AAPTS iterative method are discussed.Numerical experiments demonstrate the effectiveness of AAPTS iterative method.The sec-ond method is based on the classical Levenberg-Marquardt（LM）method,which uses the low-rank approximation of tensors-Tensor Train（TT）decomposition to overcome the high memory usage and computational complexity of tensor storage.The LMTT method for solving TAVE with TT-format is proposed by combin-ing LM method with TT decomposition.The algorithm complexity of LMTT method is linear,and it overcomes the”curse of dimensionality”phenomenon.The global convergence and local quadratic convergence of LMTT method are proved under the assumption of local error bound condition,which is weaker than the non-singularity and Lipschitz continuity conditions of Jacobian function.A hybrid method（H-LMTT）is also designed based on the TS method and the LMT-T method.Numerical experiments demonstrate the feasibility and effectiveness of these two types of LMTT（LMTT-like）methods.Finally,some numerical experi-ments are presented to verify the effectiveness of the proposed methods.