A note on unique solvability of the absolute value equation
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A note on unique solvability of the absolute value equation Shi-Liang Wu1 · Cui-Xia Li1 Received: 9 February 2019 / Accepted: 6 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this note, we show that the singular value condition σmax (B) < σmin (A) leads to the unique solvability of the absolute value equation Ax + B|x| = b for any b. This result is superior to those appeared in previously published works by Rohn (Optim Lett 3:603–606, 2009). Keywords Absolute value equation · Unique solution · Singular values
1 Introduction In this note, we consider the following absolute value equation (AVE) Ax − B|x| = b,
(1)
where A, B ∈ Rn×n and b ∈ Rn . We show that if matrices A and B satisfy σmax (B) < σmin (A), then the AVE (1) for any b has a unique solvability, where σmax and σmin , respectively, denote the maximal and minimal singular values. This result is weaker than the following condition σmax (|B|) < σmin (A), which was provided in [1] by Rohn, one can see [1] for more details.
B
Shi-Liang Wu [email protected] Cui-Xia Li [email protected]
1
School of Mathematics, Yunnan Normal University, Kunming 650500, Yunnan, People’s Republic of China
123
S.-L. Wu, C.-X. Li
At present, the AVE (1) has attracted considerable attention because the AVE (1) is used as a useful tool in optimization, such as the linear complementarity problem, linear programming and convex quadratic programming, and so on. Recently, it has been studied from two aspects: one is theoretical analysis, the other is to develop many efficient methods for solving the AVE (1). The former focuses on the theorem of alternatives, various equivalent reformulations, and the existence and nonexistence of solutions, see [1–7]. Especially, in [6], the authors presented some necessary and sufficient conditions for the unique solution of the AVE (1) with B = I , where I denotes the identity matrix. The later focuses on exploring some numerical methods for solving the AVE (1), such as the smoothing Newton method [8], the generalized Newton method [9], the sign accord method [10], the Picard-HSS method [11], the relaxed nonlinear PHSS-like method [12], Levenberg–Marquardt method [13], the finite succession of linear programs [14], the modified generalized Newton method [15,16], the preconditioned AOR method [17] and the modified Newton-type method [18].
2 The main result In this section, we will give our main result. To give our main result, the following lemma is required. Lemma 2.1 If matrices A and B satisfy σmax (B) < σmin (A), then the matrix (A − B)−1 (A + B) is positive definite. Proof Since σmax (B) < σmin (A), for all nonzero x ∈ Rn , we have x T A A T x ≥ λmin (A A T ) > λmax (B B T ) ≥ x T B B T x. Clearly, x T (A A T − B B T )x > 0. Noting that x T B A T x = x T AB T x. Further, we have 0 < x T (A A T − B B T + B A T − AB T )x = x T (A + B)(A T − B T )x.
(2)
Let (A T − B T )x = y. By the simple computations, we have x T (A + B)(A T − B T )x = y T (A − B)−1 (A + B)y. It follows that y T (A − B)−1 (A +
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