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The relaxation modulus-based matrix splitting iteration methods for circular cone nonlinear complementarity problems

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Abstract

In this paper, we study a class of nonlinear complementarity problems associated with circular cone (CCNCP for short), which is a type of non-symmetric cone complementarity problems. Useful properties of the circular cone are investigated, which help to reformulate equivalently CCNCP as an implicit fixed-point equation. Based on the implicit fixed-point equation and splittings of the system matrix, we establish a class of relaxation modulus-based matrix splitting iteration methods for solving such a complementarity problem. The convergence of the proposed modulus-based matrix splitting iteration methods has been analyzed and the strategy choice of the parameters are discussed when the splitting matrix is symmetric positive definite. Numerical experiments have shown that the modulus-based iteration methods are effective for solving CCNCP.

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References

  • Alizadeh F, Goldfarb D (2003) Second-order cone programming. Math Program 95(1):3–51

    Article  MathSciNet  MATH  Google Scholar 

  • Andersen ED, Roos C, Terlaky T (2002) Notes on duality in second order and \(p\)-order cone optimization. Optimization 51:627–643

    Article  MathSciNet  MATH  Google Scholar 

  • Andersen ED, Roos C, Terlaky T (2003) On implementing a primal-dual interior-point method for conic quadratic optimization. Math Program 95(2):249–277

    Article  MathSciNet  MATH  Google Scholar 

  • Bai Z-Z (2010) Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer Linear Algebra Appl 17:917–933

    Article  MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Zhang L-L (2013) Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer Algorithms 62(1):100–112

    Article  MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Zhang L-L (2013) Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer Algorithms 62(1):59–77

    Article  MathSciNet  MATH  Google Scholar 

  • Bonnans JF, Shapiro A (2000) Perturbation analysis of optimization problems. Springer, New York

    Book  MATH  Google Scholar 

  • Chen J-S (2006) Two classes of merit functions for the second-order cone complementarity problem. Math Methods Oper Res 64(3):495–519

    Article  MathSciNet  MATH  Google Scholar 

  • Chen J-S, Tseng P (2005) An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math Program 104(2):293–327

    Article  MathSciNet  MATH  Google Scholar 

  • Chen J-S, Pan S (2010) A one-parametric class of merit functions for the second-order cone complementarity problem. Comput Optim Appl 45(3):581–606

    Article  MathSciNet  MATH  Google Scholar 

  • Chen L-J, Ma C-F (2011) A modified smoothing and regularized Newton method for monotone second-order cone complementarity problems. Comput Math Appl 61(5):1407–1418

    Article  MathSciNet  MATH  Google Scholar 

  • Chen X-D, Sun D, Sun J (2003) Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems. Comput Optim Appl 25(1):39–56

    Article  MathSciNet  MATH  Google Scholar 

  • Dattorro J (2005) Convex optimization and Euclidean distance geometry. Meboo Publishing, Palo Alto

    MATH  Google Scholar 

  • Dong J-L, Jiang M-Q (2009) A modified modulus method for symmetric positive-definite linear complementarity problems. Numer Linear Algebra Appl 16:129–143

    Article  MathSciNet  MATH  Google Scholar 

  • Dür M (2010) Copositive programming-a survey, recent advances in optimization and its applications in engineering. Springer, Berlin

    Google Scholar 

  • Fang L (2010) A smoothing-type Newton method for second-order cone programming problems based on a new smooth function. J Appl Math Comput 34(1):147–161

    Article  MathSciNet  MATH  Google Scholar 

  • Fang L, Han C-Y (2011) A new one-step smoothing newton method for the second-order cone complementarity problem. Math Methods Appl Sci 34(3):347–359

    Article  MathSciNet  MATH  Google Scholar 

  • Faraut U, Korányi A (1994) Analysis on symmetric cones. Oxford Mathematical Monographs. Oxford University Press, New York

    MATH  Google Scholar 

  • Fukushima M, Luo Z-Q, Tseng P (2015) Smoothing functions for second-order-cone complementarity problems. SIAM J Optim 12(2):436–460

    Article  MathSciNet  MATH  Google Scholar 

  • Glineur F, Terlaky T (2004) Conic formulation for \(l_p\)-norm optimization. J Optim Theory Appl 122:285–307

    Article  MathSciNet  MATH  Google Scholar 

  • Hadjidimos A, Lapidakis M, Tzoumas M (2012) On iterative solution for linear complementarity problem with an \(H_{+}\)-matrix. SIAM J Matrix Anal Appl 33(1):97–110

    Article  MathSciNet  MATH  Google Scholar 

  • Hayashi S, Yamashita N, Fukushima M (2003) A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J Optim 15(2):593–615

    Article  MathSciNet  MATH  Google Scholar 

  • Hayashi S, Yamaguchi T, Yamashita N, Fukushima M (2005) A matrix-splitting method for symmetric affine second-order cone complementarity problems. J Comput Appl Math 175(2):335–353

    Article  MathSciNet  MATH  Google Scholar 

  • Huang N, Ma C-F (2016) The modulus-based matrix splitting algorithms for a class of weakly nonlinear complementarity problems. Numer Linear Algebra Appl 23:558–569

    Article  MathSciNet  MATH  Google Scholar 

  • Ke Y-F, Ma C-F (2014) On the convergence analysis of two-step modulus-based matrix splitting iteration method for linear complementarity problems. Appl Math Comput 243:413–418

    MathSciNet  MATH  Google Scholar 

  • Ke Y-F, Ma C-F, Zhang H (2018) The modulus-based matrix splitting iteration methods for second-order cone linear complementarity problems. Numer Algorithms. https://doi.org/10.1007/s11075-018-0484-4

    Article  MathSciNet  MATH  Google Scholar 

  • Liu S-M, Zheng H, Li W (2016) A general accelerated modulus-based matrix splitting iteration method for solving linear complementarity problems. Calcolo 53(2):189–199

    Article  MathSciNet  MATH  Google Scholar 

  • Lobo MS, Vandenberghe L, Boyd S, Lebret H (1998) Applications of second order cone programming. Linear Algebra Appl 284(1–3):193–228

    Article  MathSciNet  MATH  Google Scholar 

  • Ma C-F (2011) A regularized smoothing Newton method for solving the symmetric cone complementarity problem. Math Comput Modell 54(9–10):2515–2527

    Article  MathSciNet  MATH  Google Scholar 

  • Ma C-F, Huang N (2016) Modified modulus-based matrix splitting algorithms for a class of weakly nondifferentiable nonlinear complementarity problems. Appl Numer Math 108:116–124

    Article  MathSciNet  MATH  Google Scholar 

  • Miao X-H, Yang J-T, Hu S-L (2015) A generalized Newton method for absolute value equations associated with circular cones. Appl Math Comput 269:155–168

    MathSciNet  MATH  Google Scholar 

  • Miao X-H, Guo S-J, Qi N, Chen J-S (2016) Constructions of complementarity functions and merit functions for circular cone complementarity problem. Comput Optim Appl 63:495–522

    Article  MathSciNet  MATH  Google Scholar 

  • Monteiro RDC, Tsuchiya T (2000) Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions. Math Program 88(1):61–83

    Article  MathSciNet  MATH  Google Scholar 

  • Murty K (1988) Linear complementarity, linear and nonlinear programming. Heldermann, Berlin

    MATH  Google Scholar 

  • Narushima Y, Sagara N, Ogasawara H (2011) A smoothing newton method with Fischer-Burmeister function for second-order cone complementarity problems. J Optim Theory Appl 149(1):79–101

    Article  MathSciNet  MATH  Google Scholar 

  • Pan S, Chen J-S (2009) A damped Gauss–Newton method for the second-order cone complementarity problem. Appl Math Optim 59(3):293–318

    Article  MathSciNet  MATH  Google Scholar 

  • Schmieta SH, Alizadeh F (2001) Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones. Math Oper Res 26(3):543–564

    Article  MathSciNet  MATH  Google Scholar 

  • Schmieta SH, Alizadeh F (2003) Extension of primal-dual interior point algorithms to symmetric cones. Math Program 96(3):409–438

    Article  MathSciNet  MATH  Google Scholar 

  • Vandenberghe L, Boyd S (1996) Semidefinite programming. SIAM Rev 38:49–95

    Article  MathSciNet  MATH  Google Scholar 

  • Xia Z-C, Li C-L (2015) Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem. Appl Math Comput 271:34–42

    MathSciNet  MATH  Google Scholar 

  • Yoshise A (2005) Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cone. SIAM J Optim 17(4):1129–1153

    Article  MathSciNet  MATH  Google Scholar 

  • Zeng M-L, Zhang G-F (2015) Modulus-based GSTS iteration method for linear complementarity problems. J Math Study 48(1):1–17

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang L-L (2011) Two-step modulus-based matrix splitting iteration method for linear complementarity problems. Numer Algorithms 57(1):83–99

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang L-L, Ren Z-R (2013) Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems. Appl Math Lett 26(6):638–642

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang L-H, Yang W-H (2014) An efficient matrix splitting method for the second-order cone complementarity problem. SIAM J Optim 24:1178–1205

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang X-S, Liu S-Y, Liu Z-H (2009) A smoothing method for second order cone complementarity problem. J Comput Appl Math 228(1):83–91

    Article  MathSciNet  MATH  Google Scholar 

  • Zheng N, Yin J-F (2013) Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem. Numer Algorithms 64(2):245–262

    Article  MathSciNet  MATH  Google Scholar 

  • Zheng N, Yin J-F (2014) Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an \(H_+\)-matrix. J Comput Appl Math 260(2):281–293

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou J-C, Chen J-S (2014) On the vector-valued functions associated with circular cones. In: Abstract and applied analysis, Article ID 603542, p 21

  • Zhou J-C, Chen J-S (2013) Properties of circular cone and spectral factorization associated with circular cone. J Nonlinear Convex Anal 14(4):807–816

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by National Science Foundation of China (Nos. 41725017 and 41590864), National Key Research and Development Program of China (Nos. 2018YFC0603500, 2017YFC0404901, 2017YFC0601505 and 2017YFC0601406), National Basic Research Program of China (No. 2014CB845906), National Postdoctoral Program for Innovative Talents (No. BX201700234), and China Postdoctoral Science Foundation (No. 2017M620878). It is also partially supported by the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (No. XDB18010202).

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Authors and Affiliations

  1. College of Mathematics and Informatics and FJKLMAA, Fujian Normal University, Fuzhou, 350117, People’s Republic of China

    Yi-Fen Ke & Chang-Feng Ma

  2. Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Yi-Fen Ke & Huai Zhang

  3. Laboratory for Marine Mineral Resources, Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266237, People’s Republic of China

    Huai Zhang

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  1. Yi-Fen Ke
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  2. Chang-Feng Ma
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  3. Huai Zhang
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Correspondence to Huai Zhang.

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Communicated by Maria Aguieiras A. de Freitas.

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Ke, YF., Ma, CF. & Zhang, H. The relaxation modulus-based matrix splitting iteration methods for circular cone nonlinear complementarity problems. Comp. Appl. Math. 37, 6795–6820 (2018). https://doi.org/10.1007/s40314-018-0687-2

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