【作者】 龙梅；

【导师】 吴世良；

【作者基本信息】 云南师范大学 ， 应用数学， 2025， 硕士

【摘要】 近年来,随着张量研究领域的快速发展,垂直张量互补问题逐渐引起了学者们的关注.垂直张量互补问题不仅是张量互补问题的一种广义形式,而且与广义张量绝对值方程等价.此外,该问题源于实际应用,即当博弈中涉及的每个张量集均有限时,此类博弈可重新表述为垂直张量互补问题,并建立了广义多线性博弈的纳什均衡点与垂直张量互补问题的解之间的一一对应关系.本文旨在研究一类垂直张量互补问题的投影算法,主要分为以下几个部分:第一章介绍了一类垂直张量互补问题的背景.主要包括张量互补问题的实际应用,以及相关理论和算法研究现状.其次,系统阐述了垂直张量互补问题的现有理论研究成果及数值求解算法.第二章介绍了相关的预备知识.第三章介绍了求解一类垂直张量互补问题的投影不动点算法,其核心是将求解的问题等价转化为一个投影不动点方程,并结合幂次利普希茨张量的相关性质,得到了该算法的收敛条件.最后通过数值实验说明了所提算法的有效性.第四章在预处理技术算法论的基础上,致力于通过基于张量分裂的预处理不动点算法来求解一类垂直张量互补问题.根据涉及的特定向量的负分量数目,提出一种通用预处理子.此外,给出了所提算法的收敛性质和比较定理,并分析了参数对所提算法收敛速度的影响.第五章总结全文,并进一步提出了提高算法性能的展望.更多还原

【Abstract】 In recent years,with the rapid development of tensor research field,the vertical tensor complementarity problem(VTCP)has gradually attracted the attention of scholars.The VTCP is not only a generalized form of the tensor complementarity problem,but also is equivalent to the generalized tensor absolute value equation.Moreover,it originates from practical application,that is to say,confirmed that when each tensor set involved in the game is finite,such a game is reformulated as a VTCP and established a one-to-one correspondence between the Nash equilibrium point of the generalized multilinear game and the solution of the VTCP.This paper aims to study projected algorithms for a class of VTCP,which is divided into the following sections:Chapter One introduces the background of a class of VTCP.It mainly includes the practical applications of tensor complementarity problems and the current status of related theoretical and algorithmic research.Additionally,the existing theoretical developments and numerical solution algorithms for VTCP are systematically expounded.Chapter Two introduces the relevant preliminary knowledge.Chapter Three introduces a projected fixed point method for solving a class of VTCP.The core idea is to reformulate the problem as a projected fixed point equation,together with the relevant properties of the power Lipschitz tensor,the convergence conditions of the method are derived.Numerical experiments are given to illustrate the effectiveness of the proposed method.Chapter Four based on the methodology of the preconditioning technique,our objective is devoted to solving a class of VTCP by the preconditioned fixed point method based on tensor splitting.This chapter proposes a general precon-ditioner based on the amount of negative components of special vector involved.Furthermore,some convergence and comparison theorems of the proposed method are given and we analyze the influence of the parameter on the convergence rate of the proposed method.Chapter Five summarizes the entire paper and further proposes perspectives on improving algorithm performance.更多还原