求解无约束优化问题的两类修正的WYL共轭梯度方法
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摘要
针对无约束优化问题,通过修正共轭梯度参数,构造新的搜索方向,提出两类修正的WYL共轭梯度法.在每次迭代过程中,两类算法产生的搜索方向均满足充分下降性.在适当条件下,证明了算法的全局收敛性.数值结果表明算法是可行的和有效的.
In this paper, according to modified conjugate gradient parameter, two novel directions are developed and analyzed for classical WYL conjugate gradient method. Moreover, based on the directions, two modified WYL conjugate gradient methods are proposed for unconstrained optimization problems. Furthermore, the directions of this context are sufficient descents at each iteration. The globally convergent properties are proved under some suitable condition. Numerical results show that these methods are feasible and effective.
In this paper, according to modified conjugate gradient parameter, two novel directions are developed and analyzed for classical WYL conjugate gradient method. Moreover, based on the directions, two modified WYL conjugate gradient methods are proposed for unconstrained optimization problems. Furthermore, the directions of this context are sufficient descents at each iteration. The globally convergent properties are proved under some suitable condition. Numerical results show that these methods are feasible and effective.
引文
[1]FLETCHER R,REEVES C.Function minimization by conjugate gradients[J].Journal of Computing,1964,7:149-154.
[2]HESTENES M R,STIEFEL E L.Methods of conjugate gradients for solving linear system[J].Res.Natl.Bur.Stand.(sec.B),1952,49:409-432.
[3]POLYAK B T.The conjugate gradient method in extreme problems[J].USSR Computational Mathematic and Mathematical Physics,1969,9:94-112.
[4]POLYAK M R,RIBIERE G.Note sur la convergence des methodes de directions conjuguees[J].Rev.Francaise Imformat Recherche Operionelle,1969,16:35-43.
[5]FLETCHER R.Unconstrained Optimization Practical Methods of Optimization[M].New York:Wiley,1987.
[6]LIU Y,STOREY C.Efficient generalized conjugate gradient algorithms[J].Journal of Optimization Theory and Applications,1991,69:129-137.
[7]DAI Y H,YUAN Y.An efficient hybrid conjugate gradient method for unconstrained optimization[J].Annals Operation Research,2001,103:33-47.
[8]YUAN G,WEI Z,LU X.Global convergence of the BFGS method and the PRP method for general functions under a modified weak Wolfe-Powell line search[J].Appl.Math.Model.,2017,47:811-825.
[9]LI D H,TIAN B S.N-step quadratic convergence of the MPRP method with a restart strategy[J].J.comput.Appl.Math.,2011,235:4978-4990.
[10]ZHANG L,ZHOU W,LI D.A descent modified Polak-Ribiere-Polyak conjugate method and its global convergence[J].IMA Journal on Numerical Analysis,2006,26:629-649.
[11]YUAN G,WEI Z,LI G.A modified Polak-Ribiere-Polyak conjugate gradient algorithm for nonsmooth convex programs[J].J.Comput.Appl.Math.,2014,255:86-96.
[12]戴彧虹,袁亚湘.非线性共轭梯度法[M].上海:上海科学技术出版社,1998.
[13]YUAN G L.Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems[J].Optimization Letters,2009,3:11-21.
[14]WEI Z,YAO S,LIU L.The convergence properties of some new conjugate gradient methods[J].Applied Mathematics and Computation,2006,183:1341-1350.
[15]YAO S,WEI Z,HUANG H.A note about WYL’s conjugate gradient method and its applications[J].Applied Mathematics and Computation,2007,183:381-388.
[16]YUAN G,WEI Z,LI G.A modified Polak-Ribiere-Polyak conjugate gradient algorithm for nonsmooth convex programs[J].J.Comput.Appl.Math.,2014,255:86-96.
[17]DONG X L,LIU H W,HE Y B.A self-adjusting conjugate gradient method with sufficient descent condition and conjugacy condition[J].Journal of Optimization Theory and Applications,2015,165(1):225-241.
[18]SUN Z B,LI H Y,WANG J,et al.Two modified spectral conjugate gradient methods and their global convergence for unconstrained optimization[J].International Journal of Computer Mathematics,2018,95(10):2082-2099.
[19]DOLAN E D,MORE J J.Benchmarking optimization software with performance profiles[J].Math.Program.,Ser.A,2002,91:201-213.
[2]HESTENES M R,STIEFEL E L.Methods of conjugate gradients for solving linear system[J].Res.Natl.Bur.Stand.(sec.B),1952,49:409-432.
[3]POLYAK B T.The conjugate gradient method in extreme problems[J].USSR Computational Mathematic and Mathematical Physics,1969,9:94-112.
[4]POLYAK M R,RIBIERE G.Note sur la convergence des methodes de directions conjuguees[J].Rev.Francaise Imformat Recherche Operionelle,1969,16:35-43.
[5]FLETCHER R.Unconstrained Optimization Practical Methods of Optimization[M].New York:Wiley,1987.
[6]LIU Y,STOREY C.Efficient generalized conjugate gradient algorithms[J].Journal of Optimization Theory and Applications,1991,69:129-137.
[7]DAI Y H,YUAN Y.An efficient hybrid conjugate gradient method for unconstrained optimization[J].Annals Operation Research,2001,103:33-47.
[8]YUAN G,WEI Z,LU X.Global convergence of the BFGS method and the PRP method for general functions under a modified weak Wolfe-Powell line search[J].Appl.Math.Model.,2017,47:811-825.
[9]LI D H,TIAN B S.N-step quadratic convergence of the MPRP method with a restart strategy[J].J.comput.Appl.Math.,2011,235:4978-4990.
[10]ZHANG L,ZHOU W,LI D.A descent modified Polak-Ribiere-Polyak conjugate method and its global convergence[J].IMA Journal on Numerical Analysis,2006,26:629-649.
[11]YUAN G,WEI Z,LI G.A modified Polak-Ribiere-Polyak conjugate gradient algorithm for nonsmooth convex programs[J].J.Comput.Appl.Math.,2014,255:86-96.
[12]戴彧虹,袁亚湘.非线性共轭梯度法[M].上海:上海科学技术出版社,1998.
[13]YUAN G L.Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems[J].Optimization Letters,2009,3:11-21.
[14]WEI Z,YAO S,LIU L.The convergence properties of some new conjugate gradient methods[J].Applied Mathematics and Computation,2006,183:1341-1350.
[15]YAO S,WEI Z,HUANG H.A note about WYL’s conjugate gradient method and its applications[J].Applied Mathematics and Computation,2007,183:381-388.
[16]YUAN G,WEI Z,LI G.A modified Polak-Ribiere-Polyak conjugate gradient algorithm for nonsmooth convex programs[J].J.Comput.Appl.Math.,2014,255:86-96.
[17]DONG X L,LIU H W,HE Y B.A self-adjusting conjugate gradient method with sufficient descent condition and conjugacy condition[J].Journal of Optimization Theory and Applications,2015,165(1):225-241.
[18]SUN Z B,LI H Y,WANG J,et al.Two modified spectral conjugate gradient methods and their global convergence for unconstrained optimization[J].International Journal of Computer Mathematics,2018,95(10):2082-2099.
[19]DOLAN E D,MORE J J.Benchmarking optimization software with performance profiles[J].Math.Program.,Ser.A,2002,91:201-213.