An efficient neural network model for solving the absolute value equations (original) (raw)
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An improved generalized Newton method for absolute value equations
SpringerPlus, 2016
Background We consider the absolute value equations (AVEs): where A ∈ R n×n , b ∈ R n , and |x| denotes a vector in R n , whose i-th component is |x i |. A more general form of the AVEs, Ax + B|x| = b, was introduced by Rohn (2004) and researched in a more general context in Mangasarian (2007a). Hu et al. (2011) proposed a generalized Newton method for solving absolute value equation Ax + B|x| = b associated with second order cones, and showed that the method is globally linearly and locally quadratically convergent under suitable assumptions. As was shown in Mangasarian and Meyer (2006) by Mangasarian, the general NP-hard linear complementarity problems (LCPs) (Cottle and Dantzing 1968; Chung 1989; Cottle et al. 1992) subsume many mathematical programming problems such as absolute value equations (AVEs) (1), which own much simpler structure than any LCP. Hence it has inspired many scholars to study AVEs. And in Mangasarian and Meyer (2006) the AVEs (1) was investigated in detail theoretically, the bilinear program and the generalized LCP were prescribed there for the special case when the singular values of A are not less than 1. Based on the LCP reformulation, sufficient conditions for the existence and nonexistence of solutions are given in this paper. Mangasarian also has used concave minimization model (Mangasarian 2007b), dual complementarity (Mangasarian 2013), linear complementarity (Mangasarian 2014a), linear programming (Mangasarian 2014b) and a hybrid algorithm (Mangasarian 2015) to solve AVEs (1). Hu and Huang reformulated a system of absolute value equations as a standard linear complementarity problem without any
2012
Our goal in this work is to give an optimum correction of the infeasible absolute value equations (AVE). In order to make the mentioned system feasible, we apply the minimal correction using the l 2 norm by changing just the right hand vector. We will show that this problem can be formulated as an unconstrained optimization problem with a quadratic objective function. We propose an extension of Newton's method for solving unconstrained objective optimization. Some examples are provided to illustrate the efficiency and validity of our proposed method.