A Novel Inverse-Free Neurodynamic Approach for Solving Absolute Value Equations

We propose a novel inverse-free neurodynamic approach (NIFNA) for solving absolute value equations (AVE). The NIFNA guarantees global convergence and notably improves convergence speed by achieving fixed-time convergence. To validate the theoretical findings, numerical simulations are conducted, demonstrating the effectiveness and efficiency of the proposed NIFNA.

Concave Minimization for Sparse Solutions of Absolute Value Equations

Based on concave function, the problem of finding the sparse solution of absolute value equations is relaxed to a concave programming, and its corresponding algorithm is proposed, whose main part is solving a series of linear programming. It is proved that a sparse solution can be found under the assumption that the connected matrixes have range space property(RSP). Numerical experiments are also conducted to verify the efficiency of the proposed algorithm.

A Smoothing Neural Network Algorithm for Absolute Value Equations

In this paper, we give a smoothing neural network algorithm for absolute value equations (AVE). By using smoothing function, we reformulate the AVE as a differentiable unconstrained optimization and we establish a steep descent method to solve it. We prove the stability and the equilibrium state of the neural network to be a solution of the AVE. The numerical tests show the efficient of the proposed algorithm.

Tensor absolute value equations

This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.

Further study on tensor absolute value equations

In this paper,we consider the tensor absolute value equations(TAVEs),which is a newly introduced problem in the context of multilinear systems.Although the system of the TAVEs is an interesting generalization of matrix absolute value equations(AVEs),the well-developed theory and algorithms for the AVEs are not directly applicable to the TAVEs due to the nonlinearity(or multilinearity)of the problem under consideration.Therefore,we first study the solutions existence of some classes of the TAVEs with the help of degree theory,in addition to showing,by fixed point theory,that the system of the TAVEs has at least one solution under some checkable conditions.Then,we give a bound of solutions of the TAVEs for some special cases.To find a solution to the TAVEs,we employ the generalized Newton method and report some preliminary results.

The Sparsest Solution to the System of Absolute Value Equations

On one hand,to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas;and on the other hand,the system of absolute value equations(AVEs)has attracted a lot of attention since many practical problems can be equivalently transformed as a system of AVEs.Motivated by the development of these two aspects,we consider the problem to find the sparsest solution to the system of AVEs in this paper.We first propose the model of the concerned problem,i.e.,to find the solution to the system of AVEs with the minimum l0-norm.Since l0-norm is difficult to handle,we relax the problem into a convex optimization problem and discuss the necessary and sufficient conditions to guarantee the existence of the unique solution to the convex relaxation problem.Then,we prove that under such conditions the unique solution to the convex relaxation is exactly the sparsest solution to the system of AVEs.When the concerned system of AVEs reduces to the system of linear equations,the obtained results reduce to those given in the literature.The theoretical results obtained in this paper provide an important basis for designing numerical method to find the sparsest solution to the system of AVEs.

A New Fixed-Time Dynamical Systemfor Absolute Value Equations

A novel dynamical model with fixed-time convergence is presented to solve the system of absolute value equations(AVEs).Under a mild condition,it is proved that the solution of the proposed dynamical system converges to the solution of the AVEs.Moreover,in contrast to the existing inversion-free dynamical system(C.Chen et al.,Appl.Numer.Math.168(2021),170–181),a conservative settling-time of the proposed method is given.Numerical simulations illustrate the effectiveness of the new method.

The AOR-Base Splitting Modified Fixed Point Iteration for Solving Absolute Value Equations

Recently,Yu et al.presented a modified fixed point iterative(MFPI)method for solving large sparse absolute value equation(AVE).In this paper,we consider using accelerated overrelaxation(AOR)splitting to develop the modified fixed point iteration(denoted by MFPI-JS and MFPI-GSS)methods for solving AVE.Furthermore,the convergence analysis of the MFPI-JS and MFPI-GSS methods for AVE are also studied under suitable restrictions on the iteration parameters,and the functional equation between the parameter T and matrix Q.Finally,numerical examples show that the MFPI-JS and MFPI-GSS are efficient iteration methods.

求解绝对值方程组的广义SOR型方法

为了求解大规模的绝对值方程Ax-|x|=b,利用预处理技术及参数矩阵取代单参数的策略,文章提出了一类广义SOR型(GSOR)方法。通过选取适当的预处理矩阵或参数,GSOR方法能简化为已有的一种SOR型(NSOR)方法或导出更有效的SOR型方法。而且,基于Ax-|x|=b方程解的唯一性条件,建立了GSOR方法的收敛性定理并给出了该方法的拟最优参数。特别地,利用截断的Neumann展开构建了一个新的预处理矩阵,由此导出了一种特殊的GSOR方法,记为GSOR-1方法。文章进一步证明:GSOR-1方法具有比NSOR方法更小的拟最优收敛因子。数值测试进一步揭示:GSOR-1方法比NSOR方法具有更快的收敛速度且耗费更少的计算时间。

Absolute continuity of interacting measure-valued branching processes and its occupation-time processes

Let X t be the interaction measured_valued branching α_ symmetric stable process over R d(1<α≤2) constructed by Meleard_Roelly . Frist, it is shown that X t is absolutely continuous with respect to the Lebesgue measure (on R ) with a continuous density function which satisfies some SPDE. Second, it is proved that if the underlying process is a Brownian motion on R d (d≤3), the corresponding occupation_time process Y t is also absolutely continuous with respect to the Lebesgue measure.

求解M-矩阵绝对值方程的不动点迭代算法

多种不动点迭代方法被用于求解绝对值方程■。受此启发,本研究建立了一种求解M-矩阵绝对值方程的不动点迭代算法,在适当条件下分析了该方法的收敛性,并通过数值算例验证了该方法的可行性和有效性。

求解绝对值方程的不精确修正不动点迭代法

绝对值方程在求解线性规划、线性互补问题、双矩阵对策等问题时有着重要的意义和应用价值,本文主要研究绝对值方程的数值求解方法,为了提高计算效率,在修正的不动点迭代法的基础上,提出了一种不精确的修正不动点迭代法。在特定条件下,证明了该方法的收敛性。最后,通过数值例子验证了该方法的有效性和可行性。

求解绝对值方程的不含逆矩阵ODE方法

本文给出了一类求绝对值方程的不含逆矩阵ODE方法.先通过变量分解,将该问题转换成一个非线性方程组;然后利用Fisher价值函数,将该非线性方程组转换成一个连续可微的非线性方程;设计了一个常微分方程(ODE)来求解该非线性方程,并从理论上分析了ODE状态变量的渐进收敛性.与其他ODE方法对比,本文设计的ODE的最大特点是不需要计算与存储逆矩阵,因此其非常适合求解大规模绝对值方程.最后,初步的数值实验表明了ODE方法的有效性.

求解广义绝对值方程的两步迭代法

对于广义绝对值方程的求解问题,文章首先提出一种新的牛顿型两步迭代方法 .该方法推广了求解标准绝对值方程的一个已有两步迭代法.然后讨论新方法的收敛性,并给出一些收敛性条件.最后,通过数值算例表明本文所给方法是可行的和有效的.

波动方程all-at-once系统的快速α循环绝对值预处理

为了加快预处理MINRES方法求解波动方程all-at-once系统的收敛速度,基于绝对值预处理子和块状三对角Toeplitz预处理子,提出一种新的α循环绝对值预处理子。理论上证明了预处理矩阵可近似分裂成正交矩阵与低秩矩阵的和,且其特征值聚集在±1附近,保证了预处理MINRES方法的快速收敛性质。数值实验结果进一步表明了新预处理子的有效性。

具有2^n个解的绝对值方程问题

利用矩阵的性质,得到了绝对值方程存在2^n个解的条件,并构造了一些具有2^n个解的绝对值方程.

线性互补问题与绝对值方程的转化

给出线性互补问题与绝对值方程解存在的条件及线性互补问题与绝对值方程间的转化:包括无条件的转化和有条件的转化,并给出了线性互补问题与绝对值方程的求解方法.

改进的和声搜索算法求绝对值方程

绝对值方程Ax-|x|=b是一个不可微的NP-hard问题。给出一个改进的和声搜索算法,在种群更新过程中,采取"前拉后推"技巧来加速收敛;同时为了保持种群的多样性,在种群更新过程中以一定的概率在可行域内选取。实验结果表明,改进后的算法能够快速地求出绝对值方程尽可能多的解。

绝对值方程的一种严格可行内点算法

给出绝对值方程的一种新算法.先把绝对值方程转化为线性互补问题,再结合牛顿方向和中心路径方向,通过求解一个线性方程组得到搜索方向.获得了求解绝对值方程的一种严格可行内点算法,并证明了该算法经过有限次迭代后收敛到原问题的一个最优解,数值实验表明方法是有效的.

极大熵自适应微粒群混合算法求解绝对值方程

绝对值方程Ax-|x|=b是一个不可微的NP-hard问题。在假设矩阵A的奇异值大于1(这里矩阵A的奇异值定义为矩阵ATA特征值的非负平方根)时,给出了求解绝对值方程的一个新算法。通过引进一种极大熵函数把绝对值方程进行光滑化处理,再引入适当的目标函数,从而把绝对值方程问题转换为无约束优化问题,然后利用自适应微粒群算法对其进行求解。数值实验结果表明了该方法的正确性和有效性。