On the unique solution of a class of absolute value equations $Ax-B|Cx| = d$

1.   Introduction

At present, the absolute value equation (AVE)

and its general form

play an important role in the field of optimization (such as the complementarity problem, linear programming and convex quadratic programming), where $A, B\in \mathbb{R}^{n\times n}$. Hereby, the AVE (1.1) or (1.2) brought about widespread attention in the past several years.

Roughly speaking, the research of the AVE (1.1) or (1.2) is twofold: one, to design multifarious numerical methods for obtaining its numerical solution, see ^{[4,5,6,7,8,9,10,11,12,13]}; and, two, to present some conditions for the existence of solvability, bounds for the solutions, various equivalent reformulations, and so on, see ^{[2,3,14,15,16,17,18,19,20]}.

In ^[1], for $A, B\in \mathbb{R}^{n\times n}$, Wu found a type of new generalized absolute value equations (NGAVE) below

which is still from the linear complementarity problem. For the unique solution of the NGAVE (1.3), some necessary and sufficient conditions were presented in ^[1]. Obviously, the NGAVE (1.3) is quite other than the AVE (1.2). Whereas, if matrix $B = I$ in (1.2) and (1.3), where $I$ stands for the identity matrix, then both reduce to the AVE (1.1).

Inspired by this flurry of activities, we further generalize the concept of absolute value equation and become interested in undertaking a further study of the following absolute value equation

where $A, B, C\in \mathbb{R}^{n\times n}$. Clearly, the AVE (1.4) is a generalized form of the above three types of absolute value equations.

Similarly to the above three types of absolute value equations, the analysis of the AVE (1.4) is interesting and challenging, as a consequence of the nonlinear and nondifferentiable term $B|Cx|$ in (1.4). To our knowledge, nobody has studied the AVE (1.4) as yet. This implies that so far it is vacant for some conditions to guarantee the unique solution of the AVE (1.4). Therefore, the aim of this paper is to fill in this gap in the literature. Namely, some conditions to guarantee the unique solution of the AVE (1.4) can be gained.

2.   Main results

In this section, some conditions to guarantee the unique solution of the AVE (1.4) are presented. To achieve this goal, the following lemmas can be found in ^[1,2,3]. In the whole text, $\sigma_{1}$, $\sigma_{n}$ and $\rho$, respectively, denote the largest singular value, the smallest singular value and the spectral radius of the matrix.

Lemma 2.1. ^[1] The following statements are equal:

$(1)$ the AVE $\text{(1.3)}$ has a unique solution for any $d \in \mathbb{R}^{n}$;

$(2)$ $\mathit{\text{det}}(F_{1}(A+B)+F_{2}(A-B))\neq0$, where $F_{1}, F_{2} \in \mathbb{R}^{n\times n}$ are two arbitrary nonnegative diagonal matrices with $\mathit{\text{diag}}(F_{1}+F_{2}) > 0$;

$(3)$ $\{A+B, A- B\}$ has the row $\mathcal{W}$-property;

$(4)$ $\det(A+B)\neq0$ and $\{I, (A -B)(A+B)^{-1}\}$ has the row $\mathcal{W}$-property;

$(5)$ $\det(A+B)\neq0$ and $(A-B)(A+B)^{-1}$ is a $P$-matrix;

$(6)$ $\det(A+(I-2D)B)\neq0$ for any diagonal matrix $D = \mathit{\text{diag}}(d_{i})$ with $d_{i}\in [0, 1]$.

Lemma 2.2. ^[1] Let $\det(A)\neq0$ in $(1.3)$. When

for any diagonal matrix $D = \mathit{\text{diag}}(d_{i})$ with $d_{i}\in [0, 1]$, or

or

the AVE $(1.3)$ has a unique solution for any $d\in \mathbb{R}^{n}$.

Lemma 2.3. ^[2,3] The following statements are equal:

$(1)$ the AVE $\text{(1.2)}$ has a unique solution for any $d \in \mathbb{R}^{n}$;

$(2)$ $\mathit{\text{det}}((A-B)F_{1}+(A+B)F_{2})\neq0$, where $F_{1}, F_{2} \in \mathbb{R}^{n\times n}$ are two arbitrary nonnegative diagonal matrices with $\mathit{\text{diag}}(F_{1}+F_{2}) > 0$;

$(3)$ $\{A-B, A+B\}$ has the column $\mathcal{W}$-property;

$(4)$ $\det(A-B)\neq0$ and $\{I, (A -B)^{-1}(A+B)\}$ has the column $\mathcal{W}$-property;

$(5)$ $\det(A-B)\neq0$ and $(A-B)^{-1}(A+B)$ is a $P$-matrix;

$(6)$ $\det(A+B(I-2D))\neq0$ for any diagonal matrix $D = \mathit{\text{diag}}(d_{i})$ with $d_{i}\in [0, 1]$.

Lemma 2.4. ^[2,3] Let $\det(A)\neq0$ in $(1.2)$. When

for any diagonal matrix $D = \mathit{\text{diag}}(d_{i})$ with $d_{i}\in [0, 1]$, or

or

the AVE $(1.2)$ has a unique solution for any $d\in \mathbb{R}^{n}$.

Based on the above lemmas, naturally, we need to consider two cases for the AVE (1.4): (1) matrix $B$ is nonsingular; (2) matrix $C$ is nonsingular.

Case I. When $\det(B)\neq0$, the AVE (1.4) can be equivalently expressed as the following AVE

Based on Lemmas 2.1 and 2.2, for the AVE (1.4), we can obtain the following results, see Theorems 2.1 and 2.2.

Theorem 2.1. If $\det(B)\neq0$, then the following statements are equal:

$(1)$ the AVE $\text{(1.4)}$ has a unique solution for any $d \in \mathbb{R}^{n}$;

$(2)$ $\mathit{\text{det}}(F_{1}(B^{-1}A+C)+F_{2}(B^{-1}A-C))\neq0$, where $F_{1}, F_{2} \in \mathbb{R}^{n\times n}$ are two arbitrary nonnegative diagonal matrices with $\mathit{\text{diag}}(F_{1}+F_{2}) > 0$;

$(3)$ $\{B^{-1}A+C, B^{-1}A- C\}$ has the row $\mathcal{W}$-property;

$(4)$ $\det(B^{-1}A+C)\neq0$ and $\{I, (B^{-1}A -C)(B^{-1}A+C)^{-1}\}$ has the row $\mathcal{W}$-property;

$(5)$ $\det(B^{-1}A+C)\neq0$ and $(B^{-1}A-C)(B^{-1}A+C)^{-1}$ is a $P$-matrix;

$(6)$ $\det(A+B(I-2D)C)\neq0$ for any diagonal matrix $D = \mathit{\text{diag}}(d_{i})$ with $d_{i}\in [0, 1]$.

Theorem 2.2. Let $\det(A)\neq0$ and $\det(B)\neq0$ in $(1.4)$. When

for any diagonal matrix $D = \mathit{\text{diag}}(d_{i})$ with $d_{i}\in [0, 1]$, or

or

the AVE $(1.4)$ has a unique solution for any $d\in \mathbb{R}^{n}$.

Clearly, Theorems 2.1 and 2.2 are generalization forms of the results in Lemmas 2.1 and 2.2, respectively.

Case II. When $\det(C)\neq0$, the AVE (1.4) can be equivalently expressed as the following AVE

Based on Lemmas 2.3 and 2.4, for the AVE (1.4), we can obtain the following results, see Theorems 2.3 and 2.4.

Theorem 2.3. If $\det(C)\neq0$, then the following statements are equal:

$(1)$ the AVE $\text{(1.4)}$ has a unique solution for any $d \in \mathbb{R}^{n}$;

$(2)$ $\mathit{\text{det}}((AC^{-1}+B)F_{1}+(AC^{-1}-B)F_{2})\neq0$, where $F_{1}, F_{2} \in \mathbb{R}^{n\times n}$ are two arbitrary nonnegative diagonal matrices with $\mathit{\text{diag}}(F_{1}+F_{2}) > 0$;

$(3)$ $\{AC^{-1}-B, AC^{-1}+B\}$ has the column $\mathcal{W}$-property;

$(4)$ $\det(AC^{-1}-B)\neq0$ and $\{I, (AC^{-1} -B)^{-1}(AC^{-1}+B)\}$ has the column $\mathcal{W}$-property;

$(5)$ $\det(AC^{-1}-B)\neq0$ and $(AC^{-1}-B)^{-1}(AC^{-1}+B)$ is a $P$-matrix;

$(6)$ $\det(A+B(I-2D)C)\neq0$ for any diagonal matrix $D = \mathit{\text{diag}}(d_{i})$ with $d_{i}\in [0, 1]$.

Theorem 2.4. Let $\det(A)\neq0$ and $\det(C)\neq0$ in $(1.4)$. When

for any diagonal matrix $D = \mathit{\text{diag}}(d_{i})$ with $d_{i}\in [0, 1]$, or

or

the AVE $(1.4)$ has a unique solution for any $d\in \mathbb{R}^{n}$.

Of course, Theorems 2.3 and 2.4 are also generalization forms of the results in Lemmas 2.3 and 2.4, respectively.

In addition, based on Theorem 2.2 and Theorem 2.4, clearly, Corollary 2.1 can be otained.

Corollary 2.1. Let $\det(A)\neq0$ in $(1.4)$, and at least one of matrices $B$ and $C$ in $(1.4)$ be nonsingular. When

for any diagonal matrix $D = \mathit{\text{diag}}(d_{i})$ with $d_{i}\in [0, 1]$, or

or

the AVE $(1.4)$ has a unique solution for any $d\in \mathbb{R}^{n}$.

By investigating the condition of Corollary 2.1, to guarantee a unique solution for the AVE $(1.4)$, Corollary 2.1 not only requires the nonsingular matrix $A$, but also at least one of matrices $B$ and $C$ in $(1.4)$ is nonsingular. On this occasion, we can use the conditions (2.15), (2.16) or (2.17) to judge the unique solution of the AVE $(1.4)$.

Next, we will further relax the condition of Corollary 2.1. To this end, here, $sign(x)$ denotes the vector, which consists of $1, 0, -1$ dependent on whether $x > 0$, $x = 0$ and $x < 0$. Here, we set $y = Cx$. By making use of $|y| = Ey$ with $E = diag(sign(y))$, the AVE $(1.4)$ is equal to

which is a two-by-two block linear equation below

That is to say, we just need to establish some conditions to guarantee the unique solution of the two-by-two block linear Eq $(2.18)$.

To establish the condition of the two-by-two block linear Eq $(2.18)$, we consider the following general form of $(2.18)$, i.e.,

where $\bar{E} = diag(\bar{e_{i}})$ with any vector of components $\bar{e}_{i}\in[-1, 1]$. Clearly, the two-by-two block linear Eq $(2.18)$ is a special case of the two-by-two block linear Eq $(2.19)$. Once we give the sufficient condition for the unique solution of the two-by-two block linear Eq $(2.19)$, naturally, the sufficient condition of the unique solution of the two-by-two block linear Eq $(2.18)$ is obtained as well.

By the simple computation, we have

Based on Eq (2.20), clearly, if matrix $I-CA^{-1}B\bar{E}$ is nonsingular, then matrix

is also nonsingular. So, we know that the two-by-two block linear Eq $(2.19)$ has a unique solution. Whereupon, we have the following result for the unique solution of the AVE $(1.4)$.

Theorem 2.5. Let $\det(A)\neq0$ in $(1.4)$. When

for any diagonal matrix $\bar{E} = diag(\bar{e_{i}})$ with $\bar{e}_{i}\in[-1, 1]$, or

or

the AVE $(1.4)$ has a unique solution for any $d\in \mathbb{R}^{n}$.

Since

then Corollary 2.2 can be obtained.

Corollary 2.2. Let $\det(A)\neq0$ in $(1.4)$. If

the AVE $(1.4)$ has a unique solution for any $d\in \mathbb{R}^{n}$.

In addition, when the set of all the eigenvalues of matrix $I-CA^{-1}B\bar{E}$ does not contain 1, matrix $I-CA^{-1}B\bar{E}$ is nonsingular as well. Further, we have Theorem 2.6.

Theorem 2.6. Let $\det(A)\neq0$ in $(1.4)$. If the set of all the eigenvalues of matrix $I-CA^{-1}B\bar{E}$ does not contain 1 for any diagonal matrix $\bar{E} = diag(\bar{e_{i}})$ with $\bar{e}_{i}\in[-1, 1]$, then the AVE $(1.4)$ has a unique solution for any $d\in \mathbb{R}^{n}$.

Finally, we give a sufficient condition for the non-existence of solution of the AVE $(1.4)$, which is an extension of a non-existence result in Proposition 9 in ^[17].

Theorem 2.7. Let $B$ and $C$ be nonsingular in $(1.4)$, $0\neq B^{-1}d\geq0$, and

Then the AVE $(1.4)$ has no solution.

Proof. We prove by contradiction. Here, we assume that a nonzero solution $x$ exists. Since matrices $B$ and $C$ are nonsingular in $(1.4)$, we set $y = Cx$, then the AVE $(1.4)$ is equal to

Further, we have

So,

This is a contradiction result. This implies that the result in Theorem 2.7 holds.

Based on Theorem 2.7, we have Corollary 2.3.

Corollary 2.3. Let $B$ and $C$ be nonsingular in $(1.4)$, $0\neq B^{-1}d\geq0$, and

Then the AVE $(1.4)$ has no solution.

Clearly, Corollary 2.3 is a generalization of form of Proposition 9 in ^[17]. When $B = C = I$, Corollary 2.3 becomes Proposition 9 in ^[17].

When $C = I$ or $B = I$, naturally, we can get some sufficient conditions for the non-existence of solution of the AVE (1.2) or (1.3) by Theorem 2.7 and Corollary 2.3. Here is omitted.

3.   Conclusions

In this paper, for $A, B, C\in \mathbb{R}^{n\times n}$, we have gained some conditions to guarantee the unique solution of the absolute value equation (AVE) $Ax-B|Cx| = d$. The previous published works in ^[1,2,3] are generalized.

Acknowledgements

The authors would like to thank two anonymous referees for providing helpful suggestions, which greatly improved the paper. This research was supported by NSFC (No.11961082) and by Science and Technology Development Project of Henan Province (No. 212102210502).

Conflict of interest

The authors declare that they have no competing interests.