A modified PRP conjugate gradient method with Armijo line search for large-scale unconstrained optimization
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- 英文论文题名：A modified PRP conjugate gradient method with Armijo line search for large-scale unconstrained optimization
- 论文作者：Jianghua Yin ; Lingzhi Wang ; Xianzhen Jiang
- 英文论文作者：Jianghua Yin ; Lingzhi Wang ; Xianzhen Jiang ; College of Mathematics and Computer Science ; Guangxi Science & Technology Normal University ; College of Mathematics and Information Science ; Yulin Normal University
- 年：2017
- 作者机构：College of Mathematics and Computer Science,Guangxi Science & Technology Normal University;College of Mathematics and Information Science,Yulin Normal University;
- 英文论文关键词：Conjugate gradient method ; Sufficient descent property ; Trust region property ; Armijo line search ; Global convergence
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（B）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（B）
- 语种：英文
- 分类号：O224
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：4
- 文件大小：244k
- 原文格式：D
- 会议级别：全国
摘要
A modified PRP conjugate gradient method is proposed in this paper based on the modified secant equation. The main properties of the new method are described as follows:(i) the parameter β_k has not only gradient value information but also function value information;(ii) β_k ≥0,k;(iii) the search direction generated by the presented method possesses both the sufficient descent and trust region properties without carrying out any line search. Under some suitable assumptions, we establish the global convergence of the new method with Armijo line search. Finally, some preliminary numerical results show that the proposed method is promising and effective.
A modified PRP conjugate gradient method is proposed in this paper based on the modified secant equation. The main properties of the new method are described as follows:(i) the parameter β_k has not only gradient value information but also function value information;(ii) β_k ≥0,k;(iii) the search direction generated by the presented method possesses both the sufficient descent and trust region properties without carrying out any line search. Under some suitable assumptions, we establish the global convergence of the new method with Armijo line search. Finally, some preliminary numerical results show that the proposed method is promising and effective.
A modified PRP conjugate gradient method is proposed in this paper based on the modified secant equation. The main properties of the new method are described as follows:(i) the parameter β_k has not only gradient value information but also function value information;(ii) β_k ≥0,k;(iii) the search direction generated by the presented method possesses both the sufficient descent and trust region properties without carrying out any line search. Under some suitable assumptions, we establish the global convergence of the new method with Armijo line search. Finally, some preliminary numerical results show that the proposed method is promising and effective.
引文
[1]R.Fletcher and C.Reeves,Function minimization by conjugate gradients,The Computer Journal,7(2):149–154,1964.
[2]E.Polak and G.Ribière,Note Sur la Convergence de Directions Conjugueé,vol.16,pp.35–43.Rev.Francaise Informat Recherche Operationelle,3e Anne,1969.
[3]B.T.Polyak,The conjugate gradient method in extreme problems,USSR Computational Mathematics and Mathematical Physics,9(4):94–112,1969.
[4]M.R.Hestenes and E.L.Stiefel,Methods of conjugate gradients for solving linear systems,J.Res.Natl.Bur.Stand.49:409–436,1952.
[5]Y.Liu and C.Storey,Effcient generalized conjugate gradient algorithms part 1:theory,Journal of Optimization Theory and Applications,69(1):129–137,1991.
[6]R.Fletcher,Practical Methods of Optimization,vol.1:Unconstrained Optimization.Wiley,New York,1987.
[7]Y.H.Dai and Y.Yuan,A nonlinear conjugate gradient method with a strong global convergence property,SIAM Journal on Optimization,10(1):177–182,1999.
[8]M.J.D.Powell,Nonconvex minimization calculations and the conjugate gradient method,Lecture Notes in Mathematics,vol.1066,Spinger,Berlin,pp.122–141,1984.
[9]J.C.Gibert and J.Nocedal,Global convergence properties of conjugate gradient methods for optimization,SIAM Journal on Optimization,2(1):21–42(1992)
[10]Y.Dai,Analyses of conjugate gradient methods,PH.D.thesis,Institute of Computational Mathematics and Scientific/Engineering Computing,Chinese Academy of Sciences(in Chinese),1997.
[11]Y.Dai,and Y.Yuan.Nonlinear Conjugate Gradient Methods.Shanghai Scientific and Technical Publishers:Shanghai,People's Republic of China,1998.
[12]W.W.Hager,and H.Zhang,A new conjugate gradient method with guaranteed descent and an efficient line search,SIAM Journal on Optimization,16(1):170–192,2005.
[13]Z.Wei,S.Yao,and L.Liu,The convergence properties of some new conjugate gradient methods,Applied Mathematics and Computation,183(2):1341–1350,2006.
[14]L.Zhang,An improved Wei-Yao-Liu nonlinear conjugate gradient method for optimization computation,Applied Mathematics and computation,215(6):2269–2274,2009.
[15]G.Yuan,Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems,Optimization Letters,3(1):11–21,2009.
[16]G.Yuan and X.Lu,A modified PRP conjugate gradient method,Annals of Operations Research,166(1):73–90,2009.
[17]H.Huang and S.Lin,A modified Wei-Yao-Liu conjugate gradient method for unconstrained optimization,Applied Mathematics and Computation,231:179–186,2014.
[18]J.Jian,L.Han and X.Jiang,A hybrid conjugate gradient method with descent property for unconstrained optimization,Applied Mathematical Modelling,39(3):1281–1290,2015.
[19]Y.Cheng,Q.Mou,X.Pan,and S.Yao,A sufficient descent conjugate gradient method and its global convergence,Optimization Methods and Software,31(3):577–590,2016.
[20]X.Du,P.Zhang,and W.Ma,Some modified conjugate gradient methods for unconstrained optimization,Journal of Computational and Applied Mathematics,305:92–114,2016.
[21]S.Babaie-Kafaki,R.Ghanbari,and N.Mahdavi-Amiri,Two new conjugate gradient methods based on modified secant equations,Journal of Computational and Applied Mathematics,234(5):1374–1386,2010.
[22]N.Andrei,Accelerated hybrid conjugate gradient algorithm with modified secant condition for unconstrained optimization,Numerical Algorithms,54(1):23–46,2010.
[23]G.Yuan and M.Zhang,A modified Hestenes-Stiefel conjugate gradient algorithm for large-scale optimization,Numerical Functional Analysis and Optimization,34(8):914–937,2013.
[24]G.Yuan,Z.Wei and Q.Zhao,A modified Polak-RibièrePolyak conjugate gradient algorithm for large-scale optimization problems,IIE Transactions,46(4):397–413,2014.
[25]G.Yuan,Z.Wei and G.Li,A modified Polak-Ribière-Polyak conjugate gradient algorithm for nonsmooth convex programs,Journal of Computational and Applied Mathematics,255:86–96,2014.
[26]G.Yuan,X.Duan,W.Liu,et al.,Two new PRP conjugate gradient algorithms for minimization optimization models,Plo S one,10(10):e0140071,2015.
[27]Z.Wei,G.Li,and L.Qi,New quasi-Newton methods for unconstrained optimization problems,Applied Mathematics and Computation,175(2):1156–1188,2006.
[28]G.Yuan,and Z.Wei,Convergence analysis of a modified BFGS method on convex minimizations,Computational Optimization and Applications,47(2):237–255,2010.
[29]G.Li,C.Tang,and Z.Wei,New conjugacy condition and related new conjugate gradient methods for unconstrained optimization,Journal of Computational and Applied Mathematics,202(2):523–539,2007.
[30]M.J.D.Powell,Convergence properties of algorithm for nonlinear optimization,SIAM Review,28(4):487–500,1986.
[31]J.J.Moré,B.S.Garbow,and K.E.Hillstrom,Testing unconstrained optimization software,ACM Transactions on Mathematical Software(TOMS),7(1):17–41,1981.
[32]E.D.Dolan and J.J.Moré,Benchmarking optimization software with performance profiles,Mathematical programming,91(2):201–213,2002.
[2]E.Polak and G.Ribière,Note Sur la Convergence de Directions Conjugueé,vol.16,pp.35–43.Rev.Francaise Informat Recherche Operationelle,3e Anne,1969.
[3]B.T.Polyak,The conjugate gradient method in extreme problems,USSR Computational Mathematics and Mathematical Physics,9(4):94–112,1969.
[4]M.R.Hestenes and E.L.Stiefel,Methods of conjugate gradients for solving linear systems,J.Res.Natl.Bur.Stand.49:409–436,1952.
[5]Y.Liu and C.Storey,Effcient generalized conjugate gradient algorithms part 1:theory,Journal of Optimization Theory and Applications,69(1):129–137,1991.
[6]R.Fletcher,Practical Methods of Optimization,vol.1:Unconstrained Optimization.Wiley,New York,1987.
[7]Y.H.Dai and Y.Yuan,A nonlinear conjugate gradient method with a strong global convergence property,SIAM Journal on Optimization,10(1):177–182,1999.
[8]M.J.D.Powell,Nonconvex minimization calculations and the conjugate gradient method,Lecture Notes in Mathematics,vol.1066,Spinger,Berlin,pp.122–141,1984.
[9]J.C.Gibert and J.Nocedal,Global convergence properties of conjugate gradient methods for optimization,SIAM Journal on Optimization,2(1):21–42(1992)
[10]Y.Dai,Analyses of conjugate gradient methods,PH.D.thesis,Institute of Computational Mathematics and Scientific/Engineering Computing,Chinese Academy of Sciences(in Chinese),1997.
[11]Y.Dai,and Y.Yuan.Nonlinear Conjugate Gradient Methods.Shanghai Scientific and Technical Publishers:Shanghai,People's Republic of China,1998.
[12]W.W.Hager,and H.Zhang,A new conjugate gradient method with guaranteed descent and an efficient line search,SIAM Journal on Optimization,16(1):170–192,2005.
[13]Z.Wei,S.Yao,and L.Liu,The convergence properties of some new conjugate gradient methods,Applied Mathematics and Computation,183(2):1341–1350,2006.
[14]L.Zhang,An improved Wei-Yao-Liu nonlinear conjugate gradient method for optimization computation,Applied Mathematics and computation,215(6):2269–2274,2009.
[15]G.Yuan,Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems,Optimization Letters,3(1):11–21,2009.
[16]G.Yuan and X.Lu,A modified PRP conjugate gradient method,Annals of Operations Research,166(1):73–90,2009.
[17]H.Huang and S.Lin,A modified Wei-Yao-Liu conjugate gradient method for unconstrained optimization,Applied Mathematics and Computation,231:179–186,2014.
[18]J.Jian,L.Han and X.Jiang,A hybrid conjugate gradient method with descent property for unconstrained optimization,Applied Mathematical Modelling,39(3):1281–1290,2015.
[19]Y.Cheng,Q.Mou,X.Pan,and S.Yao,A sufficient descent conjugate gradient method and its global convergence,Optimization Methods and Software,31(3):577–590,2016.
[20]X.Du,P.Zhang,and W.Ma,Some modified conjugate gradient methods for unconstrained optimization,Journal of Computational and Applied Mathematics,305:92–114,2016.
[21]S.Babaie-Kafaki,R.Ghanbari,and N.Mahdavi-Amiri,Two new conjugate gradient methods based on modified secant equations,Journal of Computational and Applied Mathematics,234(5):1374–1386,2010.
[22]N.Andrei,Accelerated hybrid conjugate gradient algorithm with modified secant condition for unconstrained optimization,Numerical Algorithms,54(1):23–46,2010.
[23]G.Yuan and M.Zhang,A modified Hestenes-Stiefel conjugate gradient algorithm for large-scale optimization,Numerical Functional Analysis and Optimization,34(8):914–937,2013.
[24]G.Yuan,Z.Wei and Q.Zhao,A modified Polak-RibièrePolyak conjugate gradient algorithm for large-scale optimization problems,IIE Transactions,46(4):397–413,2014.
[25]G.Yuan,Z.Wei and G.Li,A modified Polak-Ribière-Polyak conjugate gradient algorithm for nonsmooth convex programs,Journal of Computational and Applied Mathematics,255:86–96,2014.
[26]G.Yuan,X.Duan,W.Liu,et al.,Two new PRP conjugate gradient algorithms for minimization optimization models,Plo S one,10(10):e0140071,2015.
[27]Z.Wei,G.Li,and L.Qi,New quasi-Newton methods for unconstrained optimization problems,Applied Mathematics and Computation,175(2):1156–1188,2006.
[28]G.Yuan,and Z.Wei,Convergence analysis of a modified BFGS method on convex minimizations,Computational Optimization and Applications,47(2):237–255,2010.
[29]G.Li,C.Tang,and Z.Wei,New conjugacy condition and related new conjugate gradient methods for unconstrained optimization,Journal of Computational and Applied Mathematics,202(2):523–539,2007.
[30]M.J.D.Powell,Convergence properties of algorithm for nonlinear optimization,SIAM Review,28(4):487–500,1986.
[31]J.J.Moré,B.S.Garbow,and K.E.Hillstrom,Testing unconstrained optimization software,ACM Transactions on Mathematical Software(TOMS),7(1):17–41,1981.
[32]E.D.Dolan and J.J.Moré,Benchmarking optimization software with performance profiles,Mathematical programming,91(2):201–213,2002.