A globally and quadratically convergent method for absolute value equations

Caccetta, Louis; Qu, Bo; Zhou, Guanglu

doi:10.1007/s10589-009-9242-9

Cited by 128 publications

(74 citation statements)

References 11 publications

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“…Results are summarized in Table 2, which gives for TAVE the number of linear programs solved, the time required to solve all the instances and gives for GN the number of linear systems and the time required. Note that other Newton methods like [2,5,20] should give similar conclusions so we do not include them in our comparison. It is to be expected that the method GN is faster than the method TAVE in particular when the dimension grows since it solves only linear systems whereas TAVE solves linear programs.…”

Section: Random Easy Problems With Unique Solutionmentioning

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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Section: Random Easy Problems With Unique Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solving absolute value equation using complementarity and smoothing functions

Abdallah

Haddou

Migot

2018

Journal of Computational and Applied Mathematics

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“…Now, we give some results about the absolute value equations and the linear complementarity problems as follows; one can see sources such as [16,29,30]. The absolute value equations have the form − | | = , where ∈ × , ∈ .…”

Section: Preliminariesmentioning

confidence: 99%

“…As we all know, the transformed linear complementarity problem is rewritten as the absolute value equation problem mainly based on the equivalence between the linear complementarity problem and the absolute value equation problem, such as [16][17][18]. The absolute value equation can be solved easily now.…”

Section: Introductionmentioning

confidence: 99%

The Smoothing FR Conjugate Gradient Method for Solving a Kind of Nonsmooth Optimization Problem with l1-Norm

Chen

2018

Mathematical Problems in Engineering

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We study the method for solving a kind of nonsmooth optimization problems with 1 -norm, which is widely used in the problem of compressed sensing, image processing, and some related optimization problems with wide application background in engineering technology. Transformated by the absolute value equations, this kind of nonsmooth optimization problem is rewritten as a general unconstrained optimization problem, and the transformed problem is solved by a smoothing FR conjugate gradient method. Finally, the numerical experiments show the effectiveness of the given smoothing FR conjugate gradient method.

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“…A smoothing Newton algorithm to solve the AVE (1) was presented by Louis Caccetta. The algorithm was proved to be globally convergent [11] and the convergence rate was quadratic under the condition that the singular values of A exceed 1. This condition was weaker than the one used in Mangasarian.…”

Section: Introductionmentioning

confidence: 99%

Improved Harmony Search Algorithm with Chaos for Absolute Value Equation

et al. 2013

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Abstrak Pada tulisan ini, pencarian harmoni plus chaos ditingkatkan (HSCH) disajikan untuk memecahkan persamaan nilai absolut (AVE) Ax -|x| = b, di mana A adalah matriks persegi sembaran dengan nilai singular melebihi satu. Hasil simulasi dalam memecahkan beberapa masalah AVE yang diberikan menunjukkan bahwa algoritma HSCH adalah valid dan lebih baik dari algoritma HS klasik (HS) dan algoritma HS dengan operator mutasi diferensial (HSDE). Kata kunci: persamaan nilai mutlak, nilai singular, harmoni pencarian, kekacauan Abstract In this paper, an improved harmony search with chaos (HSCH) is presented for solving NP-hard absolute value equation (AVE) Ax -|x| = b, where A is an arbitrary square matrix whose singular values exceed one. The simulation results in solving some given AVE problems demonstrate that the HSCH algorithm is valid and outperforms the classical HS algorithm (HS) and HS algorithm with differential mutation operator (HSDE).

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A globally and quadratically convergent method for absolute value equations

Abstract: Absolute value equations, Smoothing Newton method, Global convergence, Convergence rate,

Cited by 128 publications

References 11 publications

Solving absolute value equation using complementarity and smoothing functions

Solving absolute value equation using complementarity and smoothing functions

The Smoothing FR Conjugate Gradient Method for Solving a Kind of Nonsmooth Optimization Problem with l1-Norm

Improved Harmony Search Algorithm with Chaos for Absolute Value Equation

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