Preconditioned conjugate gradient methods for absolute value equations

Abstract

We investigate the NP-hard absolute value equations (AVE), \(Ax-B|x| =b\), where \(A,B\) are given symmetric matrices in \(\mathbb{R}^{n\times n}, \ b\in \mathbb{R}^{n}\).By reformulating the AVE as an equivalent unconstrained convex quadratic optimization, we prove that the unique solution of the AVE is the unique minimum of the corresponding quadratic optimization. Then across the latter, we adopt the preconditioned conjugate gradient methods to determining an approximate solution of the AVE.The computational results show the efficiency of these approaches in dealing with the AVE.

On unique solvability of the piecewise linear equation system

“…In this section, we introduce significant results for the unique solution of AVE (1) and GAVE (2) and establish an equivalence relation between AVE and LCP. With the help of this relation, we get some results for the AVE (1).…”

“…The study of the AVE ( 1) is challenging and interesting, as it contains non-linear and non-differential terms W |x|. Currently, many methods to solve AVE available in the literature, like the linear programming method [11], globally and quadratically convergent method [3], matrix multi-splitting Picard-iterative method [5], generalized Newton method [10], iterative method [18] and for more, one may refer to [2,13,17]. In recent years, some authors investigated the existence and uniqueness of different types of AVE (see [8,9,15,16,20,26] and references therein).…”

A note on the unique solvability condition for generalized absolute value matrix equation

“…The GAVME is a generalization form of the following generalized absolute value equation (GAVE) (2) Ax + B|x| = d, where A, B ∈ R n×n , d ∈ R n are known and x ∈ R n is unknown.…”

“…The GAVE was first introduced by Rohn [7] and studied more detail in [1,2,5,6], where authors provided its unique solvability conditions and discussed its numerical solution. The importance of absolute value equations (AVEs) is due to their broad applications in many mathematics and applied sciences domains.…”