线性圆锥互补问题的光滑化牛顿法
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摘要
给出求解线性圆锥互补问题一种新的光滑化牛顿法.首先,基于一个圆锥互补函数的光滑化函数,将线性圆锥互补问题转化成一个方程组,然后用光滑化牛顿法求解该方程组;其次,在适当假设下,证明该算法具有全局收敛性和局部二阶收敛性.数值结果表明,该算法求解线性圆锥互补问题所需的CPU时间和迭代次数均较少,且相对稳定,从而证明了算法的有效性.
We presented a new smoothing Newton method for solving the linear circular cone complementarity problems.Firstly, based on the smoothing function of the circular cone complementary function,the linear circular cone complementarity problem was transformed into a system of equations,which were solved by the smoothing Newton method.Secondly,under suitable assumptions,we proved that the algorithm had the global convergence and local quadratic convergence.The numerical results show that the CPU time and iteration times of the algorithm for solving linear circular cone complementarity problems are less,and the algorithm is relatively stable,which proves the effectiveness of the algorithm.
We presented a new smoothing Newton method for solving the linear circular cone complementarity problems.Firstly, based on the smoothing function of the circular cone complementary function,the linear circular cone complementarity problem was transformed into a system of equations,which were solved by the smoothing Newton method.Secondly,under suitable assumptions,we proved that the algorithm had the global convergence and local quadratic convergence.The numerical results show that the CPU time and iteration times of the algorithm for solving linear circular cone complementarity problems are less,and the algorithm is relatively stable,which proves the effectiveness of the algorithm.
引文
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[5]KO C H,CHEN J S.Optimal Grasping Manipulation for Multifingered Robots Using Semismooth Newton Method[J/OL].Mathematical Problems in Engineering,2013-09-11.http://dx.doi.org/10.1155/2013/681710.
[6]BOMZE I M.Copositive Optimization-Recent Developments and Applications[J].European Journal of Operational Research,2012,216(3):509-520.
[7]ZHOU Jinchuan,CHEN J S.Properties of Circular Cone and Spectral Factorization Associated with Circular Cone[J].Journal of Nonlinear and Convex Analysis,2013,14(4):807-816.
[8]BAI Yanqin,GAO Xuerui,WANG Guoqiang.Primal-Dual Interior-Point Algorithms for Convex Quadratic Circular Cone Optimization[J].Numerical Algebra,Control and Optimization,2015,5(2):211-231.
[9]DATTORRO J.Convex Optimization and Euclidean Distance Geometry[M].California:Meboo Publishing,2005.
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[11]HAYASHI S,YAMASHITA N,FUKUSHIMA M.A Combined Smoothing and Regularization Method for Monotone Second-Order Cone Complementarity Problems[J].SIAM Journal on Optimization,2005,15(2):593-615.
[12]MIAO Xinhe,GUO Shengjuan,QI Nuo,et al.Constructions of Complementarity Functions and Merit Functions for Circular Cone Complementarity Problem[J].Computational Optimization and Applications,2016,63(2):495-522.
[13]CHI Xiaoni,WAN Zhongping,ZHU Zhibin,et al.A Nonmonotone Smoothing Newton Method for Circular Cone Programming[J].Optimization,2016,65(12):2227-2250.
[14]FUKUSHIMA M,LUO Zhiquan,TSENG P.Smoothing Functions for Second-Order-Cone Complementarity Problems[J].SIAM Journal on Optimization,2002,12(2):436-460.
[15]JIANG Houyuan.Smoothed Fischer-Burmeister Equation Methods for the Complementarity Problem[J].Analog Circuits&Signal Processing,1997,153:3-4.
[2]QI Liqun,SUN Defeng,ZHOU Guanglu.A New Look at Smoothing Newton Methods for Nonlinear Complementarity Problems and Box Constrained Variational Inequalities[J].Mathematical Programming,2000,87(1):1-35.
[3]王艳,李郴良.求解线性互补问题的模系瀑布型多重网格方法[J].桂林电子科技大学学报,2016,36(2):151-153.(WANG Yan,LI Chenliang.A Modulus-Based Cascadic Multigrid Method for Linear Complementarity Problem[J].Journal of Guilin University of Electronic Technology,2016,36(2):151-153.)
[4]黄玲玲,刘三阳,王贞.求解非线性互补问题的一种新的LQP方法[J].吉林大学学报(理学版),2011,49(3):381-386.(HUANG Lingling,LIU Sanyang,WANG Zhen.A New LQP Method for Solving Nonlinear Complementarity Problems[J].Journal of Jilin University(Science Edition),2011,49(3):381-386.)
[5]KO C H,CHEN J S.Optimal Grasping Manipulation for Multifingered Robots Using Semismooth Newton Method[J/OL].Mathematical Problems in Engineering,2013-09-11.http://dx.doi.org/10.1155/2013/681710.
[6]BOMZE I M.Copositive Optimization-Recent Developments and Applications[J].European Journal of Operational Research,2012,216(3):509-520.
[7]ZHOU Jinchuan,CHEN J S.Properties of Circular Cone and Spectral Factorization Associated with Circular Cone[J].Journal of Nonlinear and Convex Analysis,2013,14(4):807-816.
[8]BAI Yanqin,GAO Xuerui,WANG Guoqiang.Primal-Dual Interior-Point Algorithms for Convex Quadratic Circular Cone Optimization[J].Numerical Algebra,Control and Optimization,2015,5(2):211-231.
[9]DATTORRO J.Convex Optimization and Euclidean Distance Geometry[M].California:Meboo Publishing,2005.
[10]ALIZADEH F,GOLDFARB D.Second-Order Cone Programming[J].Mathematical Programming,2003,95:3-51.
[11]HAYASHI S,YAMASHITA N,FUKUSHIMA M.A Combined Smoothing and Regularization Method for Monotone Second-Order Cone Complementarity Problems[J].SIAM Journal on Optimization,2005,15(2):593-615.
[12]MIAO Xinhe,GUO Shengjuan,QI Nuo,et al.Constructions of Complementarity Functions and Merit Functions for Circular Cone Complementarity Problem[J].Computational Optimization and Applications,2016,63(2):495-522.
[13]CHI Xiaoni,WAN Zhongping,ZHU Zhibin,et al.A Nonmonotone Smoothing Newton Method for Circular Cone Programming[J].Optimization,2016,65(12):2227-2250.
[14]FUKUSHIMA M,LUO Zhiquan,TSENG P.Smoothing Functions for Second-Order-Cone Complementarity Problems[J].SIAM Journal on Optimization,2002,12(2):436-460.
[15]JIANG Houyuan.Smoothed Fischer-Burmeister Equation Methods for the Complementarity Problem[J].Analog Circuits&Signal Processing,1997,153:3-4.