On the global convergence of the inexact semi-smooth Newton method for absolute value equation

Bello-Cruz, Yunier; Ferreira, O. P.; Prudente, L. F.

doi:10.1007/s10589-016-9837-x

Cited by 55 publications

(26 citation statements)

References 22 publications

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“…Using the previous lemma, problem (1) can be replaced by a sequence of concave optimization problems for r > 0 :…”

Section: Ave As a Sequence Of Concave Minimization Programsmentioning

confidence: 99%

“…In this study we put the variables in a compact set. Indeed, the functions θ r are more efficient when their arguments live in [0,1]. Besides, we use one way to express complementarity with Lemma 2.1 another way, which will be used in the numerical study, is to consider the following…”

Section: Algorithmmentioning

confidence: 99%

“…An important criterion among others is that (AVE) has a unique solution if all of the singular values of the matrix A exceed 1. In the special case where the problem is uniquely solvable, a family of Newton methods has been proposed first in [15], then completed with global and quadratic convergence in [2], an inexact version in [1] and other related methods [5,20,30]. Also, Picard-HSS iteration methods and nonlinear HSS-like methods have been considered for instance in [28,22,31].…”

Section: Introductionmentioning

confidence: 99%

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Solving absolute value equation using complementarity and smoothing functions

Abdallah

Haddou

Migot

2018

Journal of Computational and Applied Mathematics

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In this paper, we reformulate the NP-hard problem of the absolute value equation (AVE) as a horizontal linear complementarity one and then solve it using a smoothing technique. This approach leads to a new class of methods that are valid for general absolute value equation. An asymptotic analysis proves the convergence of our schemes and provides some interesting error estimates. This kind of error bound or estimate had never been studied for other known methods. The corresponding algorithms were tested on randomly generated problems and applications. These experiments show that, in the general case, one observes a reduction of the number of failures.

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“…Using the previous lemma, problem (1) can be replaced by a sequence of concave optimization problems for r > 0 :…”

Section: Ave As a Sequence Of Concave Minimization Programsmentioning

confidence: 99%

Section: Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Solving absolute value equation using complementarity and smoothing functions

Abdallah

Haddou

Migot

2018

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

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“…Saheya et al [31] focused on numerical comparisons based on four smoothing functions for (1.2). Bello Cruz et al [2] showed the global Q-linear convergence of the inexact semi-smooth Newton method for solving (1.2). Ke et al [10] studied a SOR-like iteration method for solving system of (1.2).…”

Section: Introductionmentioning

confidence: 99%

Modified HS conjugate gradient method for solving generalized absolute value equations

2019

J Inequal Appl

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We investigate a kind of generalized equations involving absolute values of variables as |A|x-|B||x| = b, where A ∈ R n×n is a symmetric matrix, B ∈ R n×n is a diagonal matrix, and b ∈ R n. A sufficient condition for unique solvability of the proposed generalized absolute value equations is also given. By utilizing an equivalence relation to the unconstrained optimization problem, we propose a modified HS conjugate gradient method to solve the transformed unconstrained optimization problem. Only under mild conditions, the global convergence of the given method is also established. Finally, the numerical results show the efficiency of the proposed method.

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“…In the last decade, some numerical methods have been developed to solve AVEs. These include the SLP method 5 , the semismooth Newton method and its inexact variants [6][7][8] , the sign accord method 9 , the hybrid algorithm 10 , the Picard-CSCS iteration method, and the nonlinear CSCSlike iteration method 11 . When A ∈ n×n is a non-Hermitian positive definite matrix, the Hermitian and skew-Hermitian splitting (HSS) iteration was first introduced by Bai et al 12 and extended in Ref.…”

Section: Introductionmentioning

confidence: 99%