一种求解对称锥互补问题的算法

系统仿真学报 ›› 2015, Vol. 27 ›› Issue (5): 1050-1056.

• 信息、控制、决策与仿真 • 上一篇下一篇

一种求解对称锥互补问题的算法

高雷阜, 于冬梅

辽宁工程技术大学优化与决策研究所, 辽宁阜新 123000

收稿日期:2014-04-15 修回日期:2014-09-24 出版日期:2015-05-08 发布日期:2020-09-01
作者简介:高雷阜(1963-),男,辽宁阜新人,教授,研究方向为最优化理论与应用、混沌动力系统预测;于冬梅(1986-),女,辽宁鞍山人,博士生,研究方向为最优化理论与应用。
基金资助:
教育部高校博士学科科研基金联合资助项目(20132121110009)

Algorithm for Solving Symmetric Cone Complementarity Problems

Gao Leifu, Yu Dongmei

Institute of Optimization and Decision, Liaoning Technical University, Fuxin 123000, China

Received:2014-04-15 Revised:2014-09-24 Online:2015-05-08 Published:2020-09-01

摘要/Abstract

摘要： 提出了一种求解对称锥互补问题的信赖域算法。基于光滑函数,将对称锥互补问题转化为无约束优化问题,构造了信赖域子问题,利用结合非单调策略的信赖域算法进行求解,并证明了该算法具有全局收敛性。仿真实验结果表明,算法对求解对称锥互补问题是有效的。

关键词: 对称锥互补问题, 信赖域算法, 非单调策略, 牛顿算法, 光滑函数

Abstract: A monotonic trust region algorithm for symmetric cone complementarity problems was proposed based on a smoothing function. The problem was transformed into unconstrained optimization problem and the trust region subproblem was constructed. The problem was solved by using trust region algorithm combined with nonmonotonic strategies, and the global convergence of the algorithm was proved. Numerical experimental results demonstrate that the algorithm is effective for symmetric cone complementarity problems.

Key words: symmetric cone complementarity problems, trust region algorithms, nonmonotonic strategies, Newton method, smooth function

中图分类号:

O2224

高雷阜, 于冬梅. 一种求解对称锥互补问题的算法[J]. 系统仿真学报, 2015, 27(5): 1050-1056.

Gao Leifu, Yu Dongmei. Algorithm for Solving Symmetric Cone Complementarity Problems[J]. Journal of System Simulation, 2015, 27(5): 1050-1056.

参考文献

[1] 修乃华, 韩继业. 对称锥互补问题[J]. 数学进展, 2007, 36(1): 1-12.
[2] Faraut J, Koranyi A.Analysis on Symmetric cones [M]. USA: Oxford University Press, 1994.
[3] Nesterov Y, Nemirovski A.Interior point polynomial methods in convex programming: heory and applications[M]. Philadelphia, USA: P Society for Industrial unapplied Mathematics (SIAM), 1994.
[4] Faybusovich L.Euclidean Jordan algebras and interior-point algorithms[J]. Positivity (S1385-1292), 1997, 1(3): 331-357.
[5] Faybusovich L.Linear systems in Jordan algebras and primal-dual interior-point algorithms[J]. J Comput Appl Math (S0377-0427), 1997, 86(3): 149-175.
[6] Schmieta S H, Alizadeh F.Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones[J]. Mathematics of Operations Research (S1432-2994), 2001, 26(03): 543-564.
[7] Schmieta S H, Alizadeh F.Extension of primal-dual interior point algorithms to symmetric cones[J]. Mathematics Programming Series A (S0025-5610), 2003, 96(03): 409-438.
[8] C B Chua, P Yi.A continuation method for nonlinear complementarity problems over symmetric cones[J]. SIAM Journal on Optimization (S0363-0129), 2010, 20(5): 2560-2583.
[9] L Fang, G P He, Y H Hu.A new smoothing Newton-type method for second-order cone programming problems[J]. Applied Mathematics and Computation (S0096-3003), 2009, 215(3): 1020-1029.
[10] Z H Huang, T Ni.Smoothing algorithms for complementarity problems over symmetric cones[J]. Comput Optim Appl (S0926-6003), 2010, 45(3): 557-579.
[11] L C Kong, J Sun, N H Xiu.A regularized smoothing Newton method for symmetric cone complementarity problems[J]. SIAM Journal on Optimization (S0363-0129), 2008, 19(3): 1028-1047.
[12] 刘长河, 刘红卫, 尚有林. 对称锥规划的邻域跟踪算法[J]. 中国科学: 数学, 2013, 43(7): 691-702.
[13] 袁亚湘, 孙文渝. 最优化理论与方法[M]. 北京: 科学出版社, 2006: 324-330.
[14] 柯小伍, 韩继业. 一类新的信赖域算法的全局收敛性[J]. 应用数学学报, 1995, 18(4): 608-615.
[15] Francisco Facchinei Christian Kanzow. A nonsmooth inexact Newton method for the solution of large-scale nonlinear Complementarity problems[J]. Mathematical Programming (S0025-5610), 1997, 76(3): 493-512.
[16] T Ni, W Z Gu.Smoothing Newton algorithm for symmetric cone complementarity problems based on a one-parametric class of smoothing functions[J]. Journal of Applied Mathematics and Computing. DOI 10.1007/s12190-009-0341-7, 2009.
[17] S P Rui, C X Xu.A smoothing inexact Newton method for nonlinear complementarity problems[J]. Journal of Computational and Applied Mathematics (S0377-0427), 2010, 233(6): 2332-2338.
[18] 张运胜, 高雷阜. 对称锥互补问题的一个惩罚NR函数的水平有界性[J]. 应用泛函分析学报, 2013, 15(4): 337-341.
[19] D Sun, J Sun.Strong semismoothness of Fischer- Burmeister SDC and SOC Methods complementarity functions[J]. Mathematical Programming (S0025-5610), 2005, 103(3): 575-581.
[20] Huang Z H, Ni T.Smoothing algorithms for complemen-
tarity problems over symmetric cones[J]. Computational Optimization and Applications (S0926-6003), 2010, 45(3): 557-579.
[21] 李改弟. 一个自动确定信赖域半径的信赖域方法[J]. 工程数学学报, 2006, 23(5): 843-848.
[22] Cottle R W, Pang J S, Stone R E.The linear complementarity problem [M]. Boston, USA: Academic Press, 1992.
[23] 王剑平, 吕毅斌, 张晓鹏. 无约束优化的一类新的非单调信赖域算法[J].科学技术与工程, 2012, 12(14): 291-294.
[24] Sturma J F.Using Sedumi 1.02, A Matlab toolbox for optimization over symmetric cones[J]. Optimization Methods and Software (S1055-6788), 1999, 11(1): 625-653.
[25] X H Liu, Z H Huang.A smoothing Newton algorithms based on one-parametric class of smoothing functions for linear programming over symmetric cones[J]. Mathematical Methods of Operations Research (S1432-2994), 2009, 70(3): 385-404.

一种求解对称锥互补问题的算法

Algorithm for Solving Symmetric Cone Complementarity Problems

PDF (PC)

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 0

编辑推荐

Metrics

本文评价