A generalized Newton method for absolute value equations associated with second order cones

Hu, Shenglong; Huang, Zheng-Hai; Zhang, Qiong

doi:10.1016/j.cam.2010.08.036

Cited by 62 publications

(44 citation statements)

References 19 publications

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“…Combining the expressions of x + (cf. (7)) and x − (cf. Lemma 2.1), we get that the expression of |x| is in the following form:…”

Section: Lemma 21mentioning

confidence: 99%

“…For the above AVEs and the CCAVEs (1), with the definition of the absolute value function over the circular cone (cf. (8)), it is easy to see that the CCAVEs (1) reduces to the SOCAVEs when θ = π 4 which be studied in [7], and the CCAVEs (1) reduces to the AVEs when n 1 = n 2 = · · · = n r = 1 and θ = π 4 . Henceforth, the CCAVEs (1) is a generalization of the previous AVEs and the SOCAVEs.…”

Section: Introductionmentioning

confidence: 96%

“…We know that the complementarity problem plays a fundamental role in optimization theory and has many applications in engineering and economics [3,4,[7][8][9]11]. Therefore, it would be worth exploring the absolute value equations, and it is one of the motivations of this work.…”

Section: Introductionmentioning

confidence: 97%

“…Recently, the AVEs have been extensively studied, see [6,7,10,13,[15][16][17][18][19] and references therein. For the above AVEs and the CCAVEs (1), with the definition of the absolute value function over the circular cone (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, it would be worth exploring the absolute value equations, and it is one of the motivations of this work. Towards to solutions of the AVEs, various numerical methods for solving the AVEs were proposed in the literature (see [2,7,13,14,25] and references therein). In this paper, we are interested in a generalized Newton method for solving the CCAVEs with m = n. More specifically, we show that the generalized Newton method is well-defined under an assumption that the all singular values of the matrix A exceed the maximal singular value of the matrix B.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A generalized Newton method for absolute value equations associated with circular cones

Miao

Yang

2015

Applied Mathematics and Computation

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“…Combining the expressions of x + (cf. (7)) and x − (cf. Lemma 2.1), we get that the expression of |x| is in the following form:…”

Section: Lemma 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A generalized Newton method for absolute value equations associated with circular cones

Miao

Yang

2015

Applied Mathematics and Computation

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Block-Diagonal and Anti-block-Diagonal Splitting Iteration Method for Absolute Value Equation

2021

Simulation Tools and Techniques

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Duality of nonconvex optimization with positively homogeneous functions

Yamanaka

Yamashita

2018

Comput Optim Appl

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We consider an optimization problem with positively homogeneous functions in its objective and constraint functions. Examples of such positively homogeneous functions include the absolute value function and the p-norm function, where p is a positive real number. The problem, which is not necessarily convex, extends the absolute value optimization proposed in [O. L. Mangasarian, Absolute value programming, Computational Optimization and Applications 36 (2007) pp. [43][44][45][46][47][48][49][50][51][52][53]. In this work, we propose a dual formulation that, differently from the Lagrangian dual approach, has a closed-form and some interesting properties. In particular, we discuss the relation between the Lagrangian duality and the one proposed here, and give some sufficient conditions under which these dual problems coincide. Finally, we show that some well-known problems, e.g., sum of norms optimization and the group Lasso-type optimization problems, can be reformulated as positively homogeneous optimization problems.Recently, the so-called absolute value equations (AVE) and absolute value optimization (AVO) problems have been attracted much attention. The AVE were introduced in 2004 by Rohn [21]. Basically, ifÃ,B are given matrices, andb is a given vector, one should find a vector x that satisfiesÃx +B|x| =b, where |x| is a vector whose i-th entry is the absolute value of the i-th entry of x. It is known that AVE are equivalent to the linear complementarity problems (LCP) [9,16,20], which include many real-world applications. As an extension of AVE, Mangasarian [14] proposed in 2007 the AVO problems, which have the absolute value of variables in their objective and constraint functions. More precisely, the AVO problem considered is given by+K|x| ≥p, whereÃ,B,H,K are given matrices, andc,d,b,p are vectors with appropriate dimensions. Since AVE and LCP are equivalent, the AVO include the mathematical programs with linear complementarity constraints [12], which are one of the formulations of equilibrium problems. As another application of AVO, Yamanaka and Fukushima [26] presented facility location problems. Since 2007, some methods for solving AVE have been presented in the literature. For example, Rohn [22] considered an iterative algorithm using the sign of variables for the case thatÃ andB are square matrices. For more generalÃ andB, Mangasarian [14] provided a method involving successive linearization techniques. Another methods include a concave minimization approach, given by Mangasarian [13], and Newton-type methods, proposed by Caccetta et al. [3], Mangasarian [15], and Zhang and Wei [28]. Some generalizations of AVE were also proposed. For example, Hu et al. [10] considered an AVE involving the absolute value of variables associated to the second-order cones. Miao et al. [18]investigated an AVE with the so-called circular cones. In both papers, quasi-Newton based algorithms were used.As for AVO problems, Yamanaka and Fukushima [26] proposed to use a branch-andbound technique. In the branching procedure, ...

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A generalized Newton method for absolute value equations associated with second order cones

Cited by 62 publications

References 19 publications

A generalized Newton method for absolute value equations associated with circular cones

A generalized Newton method for absolute value equations associated with circular cones

Block-Diagonal and Anti-block-Diagonal Splitting Iteration Method for Absolute Value Equation

Duality of nonconvex optimization with positively homogeneous functions

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