## A Dynamic Model for a Class of New Generalized Absolute Value Equations

Zheng Wenli, Tang Jia,, Chen Cairong

School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350007

 基金资助: 福建省自然科学基金面上资助项目.  2020J01166福建省自然科学基金面上资助项目.  2021J01661国家自然科学基金青年基金.  11901024

 Fund supported: the NSF of Fujian Province.  2020J01166the NSF of Fujian Province.  2021J01661the NSFC.  11901024

Abstract

In this paper, a dynamic model for solving a class of new generalized absolute value equations (GAVE) is proposed. Under suitable conditions, it can be proved that the solution of the GAVE is equivalent to the equilibrium point of the dynamic model and that the equilibrium point of the dynamic model is asymptotically stable. Numerical experiments show that the proposed dynamic model is feasible and effective.

Keywords： Generalized absolute value equation ; Linear complementarity problem ; Dynamic model ; Equilibrium point ; stability

Zheng Wenli, Tang Jia, Chen Cairong. A Dynamic Model for a Class of New Generalized Absolute Value Equations. Acta Mathematica Scientia[J], 2022, 42(3): 818-825 doi:

## 1 引言

$$$Ax-|Bx-c|=d,$$$

$$$Ax-|x|=d.$$$

AVE (1.2)也是另外两类GAVE

$$$Ax + B|x| = d$$$

$$$Ax - |Bx| = d$$$

## 2 预备知识

$$$r(x)=Q(x)-P_\Omega[Q(x)-F(x)]=Ax-|Bx-c|-d.$$$

$$$\frac{\mathrm{d}x(t)}{\mathrm{d}t}=\frac{1}{2}\gamma A^\top\{P_\Omega[Q(x(t))-F(x(t))]-Q(x(t))\},$$$

$$$\frac{\mathrm{d}x(t)}{\mathrm{d}t}=\frac{1}{2}\gamma A^\top[|Bx(t)-c|+d-Ax(t)]=h(x(t)).$$$

## 4 理论分析

$$${(x-x^{*})}^\top A^\top r(x)\geq\frac{1}{2}||r(x)||^{2}.$$$

由于$\Omega=\{x\in{{\Bbb R}} ^n|x\geq0\}\subseteq{{\Bbb R}} ^n$是闭凸子集且$Q(x^*)\in\Omega$, 则由引理2.3可得

$v:=Q(x)-F(x)$, 可以得到

$P_\Omega[\cdot]\in\Omega$, $F(x^*)\geq0 $${Q(x^*)}^\top F(x^*)=0 , 可得 利用前面得到的两个不等式和 可以得到 从而 将(2.1)式的 Q, F 的表达式代入上式可得 因为 A^\top A\succeq B^\top B , 故 定理证毕. 注4.1 若 A^\top A\succ B^\top B , 即 x^\top(A^\top A-B^\top B)x>0 , \forall 0\neq x\in{{\Bbb R}} ^n , 这等价于 x^\top (A+B)^\top(A-B)x>0 , \forall 0\neq x\in{{\Bbb R}} ^n . x=(A+B)^{-1}y , 则可以得到 y^\top (A-B)(A+B)^{-1}y>0 , 即 (A-B)(A+B)^{-1} 是正定矩阵.从而, 由注记2.1和引理2.2知此时GAVE (1.1)对 \forall c, d\in{{\Bbb R}} ^n 有唯一解.又由定理3.2知此时动力学模型(3.4)有唯一的平衡点(假设 A 非奇异). 现在, 可以分析动力学模型(3.4)的稳定性.具体地, 有如下定理. 定理4.3 假设 A 非奇异, 且 A^\top A\succeq B^\top B , 则动力学模型(3.4)的平衡点 x^* 是渐近稳定的.进一步, 如果 A^\top A\succ B^\top B , 则动力学模型(3.4)的平衡点 x^* 唯一且全局渐近稳定. 由定理4.1可知, 对于任意 x(t_0) = x_0\in {{\Bbb R}} ^n 动力学模型(3.4)在 [t_0, \infty) 有唯一解.设 x(t)=x(t;x(t_0)) 是模型(3.4)的解.构造Lyapunov函数 易知 E(x^*)=0 , 且 E(x)>0 , {\forall}x\neq x^* .由(4.1)可得 则由定理2.3可知 x^* 是渐近稳定的. A^\top A\succ B^\top B 时, 由注记4.1知动力学模型(3.4)有唯一的平衡点.因为当 ||x-x^*||\rightarrow \infty$$ E(x)\rightarrow \infty$, 由定理2.4可知此时$x^*$是全局渐近稳定的.

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