**On the asymptotic behaviour of some new gradient methods.** / Fletcher, Roger; Dai, Yu-Hong.

Research output: Contribution to journal › Article

Fletcher, R & Dai, Y-H 2005, 'On the asymptotic behaviour of some new gradient methods' *Mathematical Programming*, vol 103, no. 3, pp. 541-559. DOI: 10.1007/s10107-004-0516-9

Fletcher, R., & Dai, Y-H. (2005). On the asymptotic behaviour of some new gradient methods. *Mathematical Programming*, *103*(3), 541-559. DOI: 10.1007/s10107-004-0516-9

Fletcher R, Dai Y-H. On the asymptotic behaviour of some new gradient methods. Mathematical Programming. 2005;103(3):541-559. Available from, DOI: 10.1007/s10107-004-0516-9

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T1 - On the asymptotic behaviour of some new gradient methods

AU - Fletcher,Roger

AU - Dai,Yu-Hong

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N2 - The Barzilai-Borwein (BB) gradient method, and some other new gradient methods have shown themselves to be competitive with conjugate gradient methods for solving large dimension nonlinear unconstrained optimization problems. Little is known about the asymptotic behaviour, even when applied to n-dimensional quadratic functions, except in the case that n=2. We show in the quadratic case how it is possible to compute this asymptotic behaviour, and observe that as n increases there is a transition from superlinear to linear convergence at some value of n=4, depending on the method. By neglecting certain terms in the recurrence relations we define simplified versions of the methods, which are able to predict this transition. The simplified methods also predict that for larger values of n, the eigencomponents of the gradient vectors converge in modulus to a common value, which is a similar to a property observed to hold in the real methods. Some unusual and interesting recurrence relations are analysed in the course of the study.

AB - The Barzilai-Borwein (BB) gradient method, and some other new gradient methods have shown themselves to be competitive with conjugate gradient methods for solving large dimension nonlinear unconstrained optimization problems. Little is known about the asymptotic behaviour, even when applied to n-dimensional quadratic functions, except in the case that n=2. We show in the quadratic case how it is possible to compute this asymptotic behaviour, and observe that as n increases there is a transition from superlinear to linear convergence at some value of n=4, depending on the method. By neglecting certain terms in the recurrence relations we define simplified versions of the methods, which are able to predict this transition. The simplified methods also predict that for larger values of n, the eigencomponents of the gradient vectors converge in modulus to a common value, which is a similar to a property observed to hold in the real methods. Some unusual and interesting recurrence relations are analysed in the course of the study.

U2 - 10.1007/s10107-004-0516-9

DO - 10.1007/s10107-004-0516-9

M3 - Article

VL - 103

SP - 541

EP - 559

JO - Mathematical Programming

T2 - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

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