Convergence of the Polak–Ribiére–Polyak conjugate gradient method
In this paper, we consider the global convergence of the Polak–Ribiére–Polyak (abbreviated PRP) conjugate gradient method for unconstrained optimization problems. A new Armijo-type line search is proposed for the original PRP method and some convergence properties are given under some mild condition...
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| Published in | Nonlinear analysis Vol. 66; no. 6; pp. 1428 - 1441 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Oxford
Elsevier Ltd
15.03.2007
Elsevier |
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| Abstract | In this paper, we consider the global convergence of the Polak–Ribiére–Polyak (abbreviated PRP) conjugate gradient method for unconstrained optimization problems. A new Armijo-type line search is proposed for the original PRP method and some convergence properties are given under some mild conditions. The new Armijo-type line search can make the PRP method choose a suitable initial step size so as to decrease the function evaluations at each iteration and improve the performance of the PRP method. Numerical results show that the PRP method with the new Armijo-type line search is more efficient than other similar methods in practical computation. |
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| AbstractList | In this paper, we consider the global convergence of the Polak-Ribiere-Polyak (abbreviated PRP) conjugate gradient method for unconstrained optimization problems. A new Armijo-type line search is proposed for the original PRP method and some convergence properties are given under some mild conditions. The new Armijo-type line search can make the PRP method choose a suitable initial step size so as to decrease the function evaluations at each iteration and improve the performance of the PRP method. Numerical results show that the PRP method with the new Armijo-type line search is more efficient than other similar methods in practical computation. In this paper, we consider the global convergence of the Polak–Ribiére–Polyak (abbreviated PRP) conjugate gradient method for unconstrained optimization problems. A new Armijo-type line search is proposed for the original PRP method and some convergence properties are given under some mild conditions. The new Armijo-type line search can make the PRP method choose a suitable initial step size so as to decrease the function evaluations at each iteration and improve the performance of the PRP method. Numerical results show that the PRP method with the new Armijo-type line search is more efficient than other similar methods in practical computation. |
| Author | Shen, Jie Shi, Zhen-Jun |
| Author_xml | – sequence: 1 givenname: Zhen-Jun surname: Shi fullname: Shi, Zhen-Jun email: zjshi@qrnu.edu.cn, zjshi@umd.umich.edu, zjshi@lsec.cc.ac.cn organization: College of Operations Research and Management, Qufu Normal University, Rizhao, Shandong 276826, PR China – sequence: 2 givenname: Jie surname: Shen fullname: Shen, Jie email: shen@umich.edu organization: Department of Computer and Information Science, University of Michigan, Dearborn, MI 48128, USA |
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| Cites_doi | 10.1287/moor.3.3.244 10.1007/s10107-004-0516-9 10.1007/PL00005464 10.1007/BF02614362 10.1016/0041-5553(69)90035-4 10.1137/S1052623497318992 10.2140/pjm.1966.16.1 10.1093/comjnl/7.2.149 10.1137/0802003 10.1016/j.ejor.2003.04.003 10.1145/355934.355936 10.1007/BF00941472 10.1007/s00245-001-0003-0 10.1137/0709024 10.1017/S0962492900002270 10.1137/0715085 10.1016/j.amc.2004.10.063 10.1080/1055678042000208570 10.1137/1011036 10.1023/A:1012903105391 10.6028/jres.049.044 10.1016/j.amc.2004.06.097 |
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| Keywords | PRP conjugate gradient method Global convergence 65K05 90C30 Unconstrained optimization 49M37 Conjugate gradient method 49M37; 65K05; 90C30 Unconstrained optimization; PRP conjugate gradient method; Global convergence Nonlinear analysis |
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| References | Shi (b24) 2002; 31 Hestenes, Stiefel (b13) 1952; 49 Cohen (b3) 1972; 9 Gilbert, Nocedal (b11) 1992; 2 Nocedal, Wright (b16) 1999 Shi, Shen (b25) 2005; 167 Sun, Zhang (b20) 2001; 103 Grippo, Lucidi (b9) 2005; 20 Sun, Yang, Chen (b21) 2005; 164 Dai (b4) 2001; 89 Shi, Shen (b26) 2005; 24 Armijo (b1) 1966; 16 Moré, Garbow, Hillstrom (b15) 1981; 7 Goldstein (b12) 1965; 3 Shanno (b23) 1978; 15 Fletcher, Reeves (b8) 1964; 7 Birgin, Martinez (b2) 2001; 43 Khoda, Liu, Storey (b14) 1992; 75 Dai, Yuan (b5) 1999; 10 Dai, Fletcher (b7) 2005; 103 Nocedal (b17) 1992; 1 Grippo, Lucidi (b10) 1997; 78 Wolfe (b27) 1969; 11 Shanno (b22) 1978; 3 Yuan (b28) 1993 Dai, Ni (b6) 2003; 21 Polyak (b19) 1969; 9 Polak, Ribiére (b18) 1969; 16 Nocedal (10.1016/j.na.2006.02.001_b17) 1992; 1 Shi (10.1016/j.na.2006.02.001_b26) 2005; 24 Dai (10.1016/j.na.2006.02.001_b4) 2001; 89 Grippo (10.1016/j.na.2006.02.001_b9) 2005; 20 Armijo (10.1016/j.na.2006.02.001_b1) 1966; 16 Grippo (10.1016/j.na.2006.02.001_b10) 1997; 78 Shi (10.1016/j.na.2006.02.001_b25) 2005; 167 Nocedal (10.1016/j.na.2006.02.001_b16) 1999 Sun (10.1016/j.na.2006.02.001_b21) 2005; 164 Fletcher (10.1016/j.na.2006.02.001_b8) 1964; 7 Yuan (10.1016/j.na.2006.02.001_b28) 1993 Dai (10.1016/j.na.2006.02.001_b5) 1999; 10 Gilbert (10.1016/j.na.2006.02.001_b11) 1992; 2 Wolfe (10.1016/j.na.2006.02.001_b27) 1969; 11 Shanno (10.1016/j.na.2006.02.001_b22) 1978; 3 Cohen (10.1016/j.na.2006.02.001_b3) 1972; 9 Polyak (10.1016/j.na.2006.02.001_b19) 1969; 9 Shanno (10.1016/j.na.2006.02.001_b23) 1978; 15 Moré (10.1016/j.na.2006.02.001_b15) 1981; 7 Dai (10.1016/j.na.2006.02.001_b7) 2005; 103 Shi (10.1016/j.na.2006.02.001_b24) 2002; 31 Birgin (10.1016/j.na.2006.02.001_b2) 2001; 43 Dai (10.1016/j.na.2006.02.001_b6) 2003; 21 Goldstein (10.1016/j.na.2006.02.001_b12) 1965; 3 Khoda (10.1016/j.na.2006.02.001_b14) 1992; 75 Polak (10.1016/j.na.2006.02.001_b18) 1969; 16 Sun (10.1016/j.na.2006.02.001_b20) 2001; 103 Hestenes (10.1016/j.na.2006.02.001_b13) 1952; 49 |
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Anal. doi: 10.1137/0715085 contributor: fullname: Shanno – volume: 24 start-page: 1 issue: 3 year: 2005 ident: 10.1016/j.na.2006.02.001_b26 article-title: Step-size estimation for unconstrained optimization methods publication-title: Comput. Appl. Math. doi: 10.1016/j.amc.2004.10.063 contributor: fullname: Shi – volume: 20 start-page: 71 issue: 1 year: 2005 ident: 10.1016/j.na.2006.02.001_b9 article-title: Convergence conditions, line search algorithms and trust region implementations for the Polak Ribiére conjugate gradient method publication-title: Optim. 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Comput. doi: 10.1016/j.amc.2004.06.097 contributor: fullname: Shi – year: 1993 ident: 10.1016/j.na.2006.02.001_b28 contributor: fullname: Yuan |
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| Snippet | In this paper, we consider the global convergence of the Polak–Ribiére–Polyak (abbreviated PRP) conjugate gradient method for unconstrained optimization... In this paper, we consider the global convergence of the Polak-Ribiere-Polyak (abbreviated PRP) conjugate gradient method for unconstrained optimization... |
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| SubjectTerms | Applied sciences Calculus of variations and optimal control Exact sciences and technology Global convergence Mathematical analysis Mathematical programming Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical methods in mathematical programming Numerical methods in mathematical programming, optimization and calculus of variations Operational research and scientific management Operational research. Management science PRP conjugate gradient method Sciences and techniques of general use Unconstrained optimization |
| Title | Convergence of the Polak–Ribiére–Polyak conjugate gradient method |
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