On the unique solution of the generalized absolute value equation
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On the unique solution of the generalized absolute value equation Shiliang Wu1 · Shuqian Shen2 Received: 3 August 2020 / Accepted: 11 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, some useful necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) Ax − B|x| = b with A, B ∈ Rn×n from the optimization field are first presented, which cover the fundamental theorem for the unique solution of the linear system Ax = b with A ∈ Rn×n . Not only that, some new sufficient conditions for the unique solution of the GAVE are obtained, which are weaker than the previous published works. Keywords Generalized absolute value equation · Unique solution · Necessary and Sufficient condition Mathematics Subject Classification 90C05 · 90C30 · 65F10
1 Introduction In this paper, we concentrate on the generalized absolute value equation (GAVE), whose form is below
Ax + B|x| = b,
B
(1.1)
Shiliang Wu [email protected] Shuqian Shen [email protected]
1
School of Mathematics, Yunnan Normal University, Kunming 650500, Yunnan, People’s Republic of China
2
College of Science, China University of Petroleum, Qingdao 266580, Shandong, People’s Republic of China
123
S. Wu , S. Shen
with A, B ∈ Rn×n and b ∈ Rn . When B = I , where I stands for the identity matrix, the GAVE (1.1) reduces to the absolute value equation (AVE) Ax + |x| = b.
(1.2)
The GAVE (AVE) have received considerable attention because they are used as a useful tool in the optimization field, such as the complementarity problem, linear programming and convex quadratic programming, see [1–5]. Especially, for solving the well-known linear complementarity problem (LCP), the LCP is to find z such that w = M z + q ≥ 0, z ≥ 0 and z T w = 0, with M ∈ Rn×n , q ∈ Rn ,
(1.3)
many efficient numerical methods can be established on the base of the form of the GAVE (1.1), such that the modulus-type iteration method [6,7] and its various versions [8,9], the generalized Newton method [10], and so on. The research of the unique solution is a very important branch of theoretical analysis of the GAVE (AVE). By observing the structure of the GAVE (1.1), it is not difficult to see that the nonlinear and nondifferentiable term B|x| often leads to the nondeterminacy of the solution of the GAVE (1.1). In this case, we have to give some constraints to guarantee that the GAVE (1.1) has a unique solution. Recently, some sufficient conditions for the unique solution of the GAVE (1.1) have been obtained in the literatures. For example, in [13], Rohn et al. found that the GAVE (1.1) for any b ∈ Rn has a unique solution if ρ(|A−1 B|) < 1, where ρ(·) denotes the spectral radius of the matrix. From the singular value of the matrix, Rohn in [14] showed that the GAVE (1.1) for any b ∈ Rn has a unique solution when σ1 (|B|) < σn (A), where σ1 and σn , respectively, denote the maximal and minimal singular value of the matrix. Based on the work in [14], Wu and Li in [15] obtained an improved res
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