求解二阶锥绝对值方程组的非单调光滑牛顿算法

doi:10.12286/jssx.j2022-0980

计算数学 ›› 2023, Vol. 45 ›› Issue (2): 251-266.DOI: 10.12286/jssx.j2022-0980

• 论文 • 上一篇

求解二阶锥绝对值方程组的非单调光滑牛顿算法

于冬梅¹, 王增伟¹, 陈彩荣², 韩德仁³

1. 辽宁工程技术大学理学院, 阜新 123000;
2. 福建师范大学数学与统计学院, 福州 350117;
3. 北京航空航天大学数学科学学院, 信息与行为教育部重点实验室, 北京 100191

收稿日期:2022-07-28 出版日期:2023-05-14 发布日期:2023-05-13
通讯作者: 陈彩荣, Email:cairongchen@fjnu.edu.cn
基金资助:
国家自然科学基金 (12201275, 11901024, 12131004), 国家重点研发计划 (2021YFA1003600), 辽宁省教育厅基金 (LJ2019JL017, LJ2019JL005) 和福建省自然科学基金 (2021J01661) 资助.

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于冬梅, 王增伟, 陈彩荣, 韩德仁. 求解二阶锥绝对值方程组的非单调光滑牛顿算法[J]. 计算数学, 2023, 45(2): 251-266.

Yu Dongmei, Wang Zengwei, Chen Cairong, Han Deren. A NON-MONOTONE SMOOTHING NEWTON ALGORITHM FOR ABSOLUTE VALUE EQUATIONS ASSOCIATED WITH SECOND-ORDER CONE[J]. Mathematica Numerica Sinica, 2023, 45(2): 251-266.

A NON-MONOTONE SMOOTHING NEWTON ALGORITHM FOR ABSOLUTE VALUE EQUATIONS ASSOCIATED WITH SECOND-ORDER CONE

Yu Dongmei¹, Wang Zengwei¹, Chen Cairong², Han Deren³

1. College of Science, Liaoning Technical University, Fuxin 123000, China;
2. School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China;
3. School of Mathematical Sciences, Beihang University, LMIB of the Ministry of Education, Beijing 100191, China

Received:2022-07-28 Online:2023-05-14 Published:2023-05-13

本文提出了求解二阶锥绝对值方程组 (SOCAVE) 的非单调光滑牛顿算法. 在适当的条件下分析了算法的全局收敛性和局部二次收敛性. 数值结果表明用非单调光滑牛顿算法求解 SOCAVE 是可行且高效的.

关键词： 绝对值方程组, 二阶锥, 光滑牛顿算法, 非单调线搜索

A non-monotone smoothing Newton method is proposed to solve the system of absolute value equations associated with second-order cone. Under certain conditions, we prove that the proposed method is globally and locally quadratically convergent. Numerical results verify that the proposed method can efficiently solve the system of absolute value equations associated with second-order cone.

Key words: absolute value equation, second-order cone, smoothing Newton algorithm, non-monotone line search

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[1] Alizadeh F, Goldfarb D. Second-order cone programming[J]. Mathematical Programming, 2003, 95:3-51.
[2] Chen J S, Pan S H. A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs[J]. Pacific Journal of Optimization, 2012, 8(1):33-74.
[3] Chen C R, Yu D M, Han D R. Exact and inexact Douglas-Rachford splitting methods for solving large-scale sparse absolute value equations[J]. IMA Journal of Numerical Analysis, 2022. https://doi.org/10.1093/imanum/drab105.
[4] Chen C R, Yang Y N, Yu D M, Han D R. An inverse-free dynamical system for solving the absolute value equations[J]. Applied Numerical Mathematics, 2021, 168:170-181.
[5] Dong X, Shao X H, Shen H L. A new SOR-like method for solving absolute value equations[J]. Applied Numerical Mathematics, 2020, 156:410-421.
[6] Faraut J, Korányi A. Analysis on Symmetric Cones[M]. New York:Oxford University Press, 1994.
[7] Fischer A. A special Newton-type optimization method[J]. Optimization, 1992, 24:269-284.
[8] Fukushima M, Luo Z Q, Tseng P. Smoothing functions for second-order-cone complementarity problems[J]. SIAM Journal on Optimization, 2002, 12(2):436-460.
[9] Hu S L, Huang Z H, Chen J S. Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems[J]. Journal of Computational and Applied Mathematics, 2009, 230:69-82.
[10] Hu S L, Huang Z H, Zhang Q. A generalized Newton method for absolute value equations associated with second order cones[J]. Journal of Computational and Applied Mathematics, 2011, 235:1490-1501.
[11] Huang Z H, Ni T. Smoothing algorithms for complementarity problems over symmetric cones[J]. Computational Optimization and Applications, 2010, 45:557-579.
[12] Ke Y F, Ma C F. SOR-like iteration method for solving absolute value equations[J]. Applied Mathematics and Computation, 2017, 311:195-202.
[13] Ke Y F. The new iteration algorithm for absolute value equation[J]. Applied Mathematics Letters, 2020, 99:105990.
[14] Ju X X, Li C D, Han X, He X. Neurodynamic network for absolute value equations:A fixed-time convergence technique[J]. IEEE Transactions on Circuits and Systems II-Express Briefs, 2022, 69(3):1807-1811.
[15] Mangasarian O L. Absolute value programming[J]. Computational Optimization and Applications, 2007, 36:43-53.
[16] Mangasarian O L. Knapsack feasibility as an absolute value equation solvable by successive linear programming[J]. Optimization Letters, 2009, 3:161-170.
[17] Mangasarian O L. A generalized Newton method for absolute value equations[J]. Optimization Letters, 2009, 3:101-108.
[18] Mangasarian O L, Meyer R R. Absolute value equations[J]. Linear Algebra and its Applications, 2006, 419:359-367.
[19] Mezzadri F. On the solution of general absolute value equations[J]. Applied Mathematics Letters, 2020, 107:106462.
[20] Miao X H, Yang J T, Saheya B, Chen J S. A smoothing Newton method for absolute value equation associated with second-order cone[J]. Applied Numerical Mathematics, 2017, 120:82-96.
[21] Miao X H, Hsu W M, Nguyen C T, Chen J S. The solvabilities of three optimization problems associated with second-order cone[J]. Journal of Nonlinear and Convex Analysis, 2021, 22(5):937-967.
[22] Nguyen C T, Saheya B, Chang Y L, Chen J S. Unified smoothing functions for absolute value equation associated with second-order cone[J]. Applied Numerical Mathematics, 2019, 135:206-227.
[23] Prokopyev O. On equivalent reformulations for absolute value equations[J]. Computational Optimization and Applications, 2009, 44:363-372.
[24] Qi L Q. Convergence analysis of some algorithms for solving nonsmooth equations[J]. Mathematics of Operations Research, 1993, 18(1):227-244.
[25] Qi L Q, Sun J. A nonsmooth version of Newton's method[J]. Mathematical Programming, 1993, 58:353-367.
[26] Qi L Q, Sun D F, Zhou G L. A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities[J]. Mathematical Programming, 2000, 87:1-35.
[27] Hladík M, Moosaei H. Some notes on the solvability conditions for absolute value equations[J]. Optimization Letters, 2022. https://doi.org/10.1007/s11590-022-01900-x.
[28] Tang J Y, Zhou J C. Quadratic convergence analysis of a nonmonotone Levenberg-Marquardt type method for the weighted nonlinear complementarity problem[J]. Computational Optimization and Applications, 2021, 80:213-244.
[29] Tang J Y, Zhang H C. A nonmonotone smoothing Newton algorithm for weighted complementarity problem[J]. Journal of Optimization Theory and Applications, 2021, 189:679-715.
[30] Wang A, Cao Y, Chen J X. Modified Newton-type iteration methods for generalized absolute value equations[J]. Journal of Optimization Theory and Applications, 2019, 181:216-230.
[31] Yu D M, Chen C R, Han D R. A modified fixed point iteration method for solving the system of absolute value equations[J]. Optimization, 2022, 71(3):449-461.
[32] Yu D M, Chen C R, Yang Y N, Han D R. An inertial inverse-free dynamical system for solving absolute value equations[J]. Journal of Industrial and Management Optimization, 2022. https://doi.org/10.3934/jimo.2022055.
[33] Zhang H C, Hager W W. A nonmonotone line search technique and its application to unconstrained optimization[J]. SIAM Journal on Optimization, 2004, 14(4):1043-1056.
[34] Zhang J L, Zhang G F, Liang Z Z. A modified generalized SOR-like method for solving an absolute value equation[J]. Linear and Multilinear Algebra, 2022. https://doi.org/10.1080/03081087.2022.2066614.
[35] Zamani M, Hladík M. Error bounds and a condition number for the absolute value equations[J]. Mathematical Programming, 2022. https://doi.org/10.1007/s10107-021-01756-6.

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摘要

求解二阶锥绝对值方程组的非单调光滑牛顿算法

A NON-MONOTONE SMOOTHING NEWTON ALGORITHM FOR ABSOLUTE VALUE EQUATIONS ASSOCIATED WITH SECOND-ORDER CONE

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