A fixed-time inverse-free dynamical system for solving the system of absolute value equations

To solve the system of absolute value equations (AVEs) is to find an $x\in\mathbb{R}^{n}$ such that

$$Ax-|x|-b=0,$$

(1.1)

where $A\in\mathbb{R}^{n\times n}$, $b\in\mathbb{R}^{n}$ and $|x|$ represents the componentwise absolute value of the unknown vector $x$. The AVEs (1.1) is a special case of the generalized absolute value equations (GAVEs)

$$Cx-D|x|-c=0,$$

(1.2)

where $C,\,D\in\mathbb{R}^{m\times n}$, $x\in\mathbb{R}^{n}$ and $c\in\mathbb{R}^{m}$. The GAVEs (1.2) is originally introduced by Rohn in [23] and further investigated in [14, 21, 6] and the references therein. Over the past two decades, the AVEs (1.1) has been widely studied in the optimization community because of its relevance to many mathematical programming problems, such as the linear complementarity problem (LCP) [21, 14, 16, 8], the horizontal LCP (HLCP) [19], the generalized LCP (GLCP) [16] and others, see e.g. [21, 16, 15] and the references therein. In addition, the AVEs (1.1) is closely related to the system of linear interval equations [22].

In general, it has been shown in [14] that solving the GAVEs (1.2) is NP-hard. Moreover, if the GAVEs (1.2) is solvable, it follows from [21] that checking whether the GAVEs (1.2) has a unique solution or multiple solutions is NP-complete. Throughout this paper, we assume that $A$ is invertible and $\|A^{-1}\|<1$, and thus the AVEs (1.1) has a unique solution for any $b\in\mathbb{R}^{n}$ [16]. For more discussions about the unique solvability of the AVEs (1.1), see e.g. [28, 19, 24, 27, 7] and the references therein.

In this paper, we focus on further studying the continuous solution schemes for solving the AVEs (1.1). We briefly introduce some existing works in the following. To this end, we recall two reformulations of the AVEs (1.1).

The AVEs (1.1) is equivalent with the following GLCP [16]:

$$Q(x)\doteq Ax+x-b\geq 0,\quad F(x)\doteq Ax-x-b\geq 0\quad\hbox{and}\quad\left\langle Q(x),F(x)\right\rangle=0.$$

(1.3)

Furthermore, if $1$ is not an eigenvalue of $A$, then the AVEs (1.1) can be reformulated as the following LCP [16]:

$$u\geq 0,\quad(A+I)(A-I)^{-1}u+q\geq 0\quad\hbox{and}\quad\left\langle u,(A+I)(A-I)^{-1}u+q\right\rangle=0$$

(1.4)

with

$$q=\left[(A+I)(A-I)^{-1}-I\right]b,\quad u=(A-I)x-b.$$

(1.5)

It follows from (1.4) and (1.5) that if $u^{*}$ is a solution of the LCP (1.4), then $x^{*}=(A-I)^{-1}(u^{*}+b)$ is a solution of the AVEs (1.1).

By utilizing the reformulation (1.4) with (1.5) of the AVEs (1.1), some dynamical systems are constructed to solve the AVEs (1.1). For instance, the following dynamical model

		$\displaystyle\text{state~{}equation:}\quad\frac{du}{dt}=P_{\Omega}[u-\lambda g(u,\beta)]-u,$
		$\displaystyle\text{output~{}equation:}\quad x=(A-I)^{-1}(u+b)$

is used by Mansoori, Eshaghnezhad and Effati [18] to solve the AVEs (1.1), where $0<\lambda\leq 1$, $0<\beta<\frac{1}{5L}$, and

	$\displaystyle g(u,\beta)=e(u,\beta)-\beta Me(u,\beta),$
	$\displaystyle e(u,\beta)=u-P_{\Omega}[u-\beta(Mu+q)],$
	$\displaystyle L=\|M\|,\;M=(A+I)(A-I)^{-1},$
	$\displaystyle\Omega=\{x\in\mathbb{R}^{n}|x\geq 0\}.$

Huang and Cui [10] use the following dynamical system

	$\displaystyle\text{state~{}equation:}\quad\frac{du}{dt}=-\gamma e(u,1),$
	$\displaystyle\text{output~{}equation:}\quad x=(A-I)^{-1}(u+b)$

to solve the AVEs (1.1) and Mansoori and Erfanian [17] suggest the following dynamical system

	$\displaystyle\text{state~{}equation:}\quad\frac{du}{dt}=-\gamma(I+M^{\top})e(u,1),$
	$\displaystyle\text{output~{}equation:}\quad x=(A-I)^{-1}(u+b)$

to solve the AVEs (1.1), where $\gamma>0$ is the convergence rate. Recently, Ju, Li, Han and He [11] develop a fixed-time dynamical system for solving the AVEs (1.1). Based on the relation with HLCP, the following dynamical system is proposed in [4] to solve AVEs (1.1):

	$\displaystyle\text{state~{}equation:}\quad\frac{dz}{dt}=\frac{\rho}{2}(|x|-z),$
	$\displaystyle\text{output~{}equation:}\quad x=A^{-1}(z+b),$

where $\rho>0$ is a scaling constant. Obviously, all of the above mentioned dynamical systems involve the inversion of the matrix $A-I$ or $A$. In order to avoid the inversion, based on (1.3), Chen, Yang, Yu and Han [2] develop the following inverse-free dynamical system

$$\frac{dx}{dt}=\gamma A^{\top}(b+|x|-Ax)\doteq-\gamma A^{\top}r(x)\doteq g(\gamma,x),$$

(1.6)

where $\gamma>0$ is the convergence rate parameter. The dynamical system (1.6) can trace back to [29, 13, 9, 3]. Subsequently, an inertial version of (1.6) is developed in [31]. Other inverse-free dynamical system appears in [25, 26, 30], in which the smoothing technique is used.

It is known that the finite-time convergence proposed in [1] is of practical interests than the classic asymptotical stability or exponential stability over infinite time [11]. To overcome the limitation that the settling-time of the finite-time model is initial condition dependent, the concept of the so-called fixed-time convergence [20] is developed. Inspired by the works of [2, 11], we will develop a fixed-time inverse-free dynamical system for solving the AVEs (1.1). The main features of our method can be summarized as follows.

• (a)

  Comparing with the methods proposed in [18, 10, 17, 4], the proposed method is inverse-free and possesses a conservative settling-time.

• (b)

  Comparing with the method proposed in [11], the proposed method is inverse-free.

• (c)

  Comparing with the methods proposed in [2, 31], the proposed method has a conservative settling-time.

• (d)

  Comparing with the methods proposed in [25, 26, 30], the proposed method has a conservative settling-time and does not need any smoothing function.

The rest of this paper is organized as follows. In section 2 we state a few basic results on the AVEs (1.1) and the autonomous system, relevant to our later developments. The fixed-time inverse-free dynamical system to solve the AVEs (1.1) is developed in section 3 and its convergence analysis is also given there. Numerical simulations are given in section 4. Conclusions are made in section 5.

Notation. We use $\mathbb{R}^{n\times n}$ to denote the set of all $n\times n$ real matrices and $\mathbb{R}^{n}=\mathbb{R}^{n\times 1}$. We use $\mathbb{R}_{+}$ to denote the nonnegative reals. $I$ is the identity matrix with suitable dimension. $|\cdot|$ denotes absolute value for real scalar. The transposition of a matrix or vector is denoted by $\cdot^{\top}$. The inner product of two vectors in $\mathbb{R}^{n}$ is defined as $\langle x,y\rangle\doteq x^{\top}y=\sum\limits_{i=1}^{n}x_{i}y_{i}$ and $\|x\|\doteq\sqrt{\langle x,x\rangle}$ denotes the $2$-norm of vector $x\in\mathbb{R}^{n}$. $\|A\|$ denotes the spectral norm of $A$ and is defined by the formula $\|A\|\doteq\max\left\{\|Ax\|:x\in\mathbb{R}^{n},\|x\|=1\right\}$. The smallest singular value and the smallest eigenvalue of $A$ are denoted by $\sigma_{\min}(A)$ and $\lambda_{\min}(A)$, respectively. $\textbf{tridiag}(a,b,c)$ denotes a matrix that has $a,b,c$ as the subdiagonal, main diagonal and superdiagonal entries in the matrix, respectively. The projection mapping from $\mathbb{R}^{n}$ onto $\Omega$, denoted by $P_{\Omega}$, is defined as $P_{\Omega}[x]=\arg\min\{\|x-y\|:y\in\Omega\}$.

In this section, we collect a few important results on the autonomous system and the AVEs (1.1), which lay the foundation of our later arguments.

Before talking something about the autonomous system, we give the definition of Lipschitz continuity.

Consider the autonomous system

$$\frac{dx}{dt}=f(x(t)),\quad x(0)=x_{0}\in\mathbb{R}^{n},$$

(2.1)

where $f$ is a function from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}$. Throughout this paper, denote $x(t;x_{0})$ the solution of (2.1) determined by the initial value condition $x(0)=x_{0}$.

Before ending this section, we will give some properties of the AVEs (1.1).

In this section, inspired by the works of [2, 11], we present a fixed-time inverse-free dynamical system for solving the AVEs (1.1).

The developed fixed-time inverse-free dynamical system is as follows:

$$\begin{array}[]{l}\frac{dx}{dt}=\rho(x)g(\gamma,x),\\ \rho(x)=\begin{cases}\frac{\rho_{1}}{\|r(x)\|^{1-\lambda_{1}}}+\frac{\rho_{2}}{\|r(x)\|^{1-\lambda_{2}}},&\text{if}\quad x\notin\boldsymbol{Fix}(r),\\ 0,&\text{if}\quad x\in\boldsymbol{Fix}(r),\end{cases}\end{array}$$

(3.1)

where $\rho_{1},\,\rho_{2},\gamma>0$, $\lambda_{1}\in(0,1)$, $\lambda_{2}\in(1,+\infty)$ and $\boldsymbol{Fix}(r)=\{x\in\mathbb{R}^{n}:r(x)=0\}$.

The following theorem follows from Lemma 2.1 and Lemma 3.1.

Now we are in the position to explore the convergence of the proposed model (3.1).

In this section, we will present one example to illustrate the effectiveness of the proposed method. All experiments are implemented in MATLAB R2018b with a machine precision $2.22\times 10^{-16}$ on a PC Windows 10 operating system with an Intel i7-9700 CPU and 8GB RAM. The ODE solver used is “ode23”. Concretely, the MATLAB expression

$$[t,y]=\text{ode23}(odefun,tspan,y_{0})$$

is used, which integrates the system of differential equations from $t_{0}$ to $t_{f}$ with $tspan=[t_{0},t_{f}]$. Here, “odefun” is a function handle.

In this paper, a fixed-time dynamical system is proposed to solve the AVEs (1.1). Theoretical results show that the presented model is globally convergent and has a conservative settling-time. Numerical comparison with the method proposed in [11] shows that our method is preferred, at least in the sense that our method is inverse-free.