Absolute value equations

Mangasarian, O. L.; Meyer, Robert R.

doi:10.1016/j.laa.2006.05.004

Cited by 275 publications

(200 citation statements)

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“…In [2,3,16], many mathematical programming problems can be reduced to a linear complementary (LCP) and LCP is equivalent to an absolute value equation (1.1). And many methods are proposed for solving AVE (see [5-15, 17, 19-24, 26]).…”

Section: Introductionmentioning

confidence: 99%

Picard splitting method and Picard CG method for solving the absolute value equation

Lv¹,

Ma²

2017

J. Nonlinear Sci. Appl.

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Section: Introductionmentioning

confidence: 99%

Picard splitting method and Picard CG method for solving the absolute value equation

Lv¹,

Ma²

2017

J. Nonlinear Sci. Appl.

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“…John [1] provided a theorem of the alternatives for a more general form of AVE (1), Ax B x b   , and enlightens the relation between the AVE (1) and the interval matrix. The AVE (1) is shown to be equivalent to the bilinear program, the generalized LCP, and the standard LCP if 1 is not an eigenvalue of A by Mangasarian [4].…”

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 slightly more general form of the AVE, Ax + B|x| = b was introduced in [14] that was investigated algorithmically in [15] and in a more general context in [8]. The AVE (1.1) was investigated in detail theoretically in [11], and a bilinear program in the primal space of the problem was prescribed there for the special case when the singular values of A are not less than one. No computational results were given in either [11] or [8].…”

Section: Introductionmentioning

confidence: 99%

“…The AVE (1.1) was investigated in detail theoretically in [11], and a bilinear program in the primal space of the problem was prescribed there for the special case when the singular values of A are not less than one. No computational results were given in either [11] or [8]. In contrast in [7], computational results were given for a linear-programming-based successive linearization algorithm utilizing a concave minimization model and a primal-dual bilinear model in [10].…”

Section: Introductionmentioning

confidence: 99%

“…In contrast in [7], computational results were given for a linear-programming-based successive linearization algorithm utilizing a concave minimization model and a primal-dual bilinear model in [10]. As was shown in [11], the general NP-hard linear complementarity problem (LCP) [2,3,1], which subsumes many mathematical programming problems, can be formulated as an AVE (1.1). This implies that (1.1) is NP-hard in its general form.…”

Section: Introductionmentioning

confidence: 99%

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