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Globally convergent conjugate gradient algorithms. (English) Zbl 0579.90079
The paper deals with the convergence of the conjugate gradient algorithms for unconstrained minimization. A test is proposed, determining when to restart the conjugate gradient methods by using the steepest descent search direction. The test examines the angle between the search direction and the negative gradient and guarantees convergence of the algorithm to a stationary point of the objective function when the latter is twice continuously differentiable and has a bounded level set.
Reviewer: V.M.Veliov

MSC:
90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
49M37 Numerical methods based on nonlinear programming
Software:
Algorithm 500
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References:
[1] E.M.L. Beale, ”A derivation of conjugate gradients”, in: F.A. Lootsma, ed.,Numerical methods for nonlinear optimization (Academic Press, London, 1972) pp. 39–43.
[2] R. Fletcher and C.M. Reeves, ”Function minimization by conjugate gradients”,Computer Journal 7 (1964) 149–154. · Zbl 0132.11701
[3] M.R. Hestenes and E. Stiefel, ”Methods of conjugate gradients for solving linear systems”,Journal of Research of the National Bureau of Standards, Sec. B 48 (1952) 409–436. · Zbl 0048.09901
[4] J.J. Moré, B.S. Garbow and K.E. Hillstrom, ”Testing unconstrained minimization software”, TM-324, Applied Mathematics Division, Argonne National Laboratory, Argonne, IL (1978). · Zbl 0454.65049
[5] E. Polak and G. Ribière, ”Note sur la convergence de méthodes de directions conjuguées”, RAIRO 16 (1969) 35–43. · Zbl 0174.48001
[6] M.J.D. Powell, ”Restart procedures for the conjugate gradient method,Mathematical Programming 12 (1977) 241–254. · Zbl 0396.90072
[7] M.J.D. Powell, ”Nonconvex minimization calculations and the conjugate gradient method”, DAMTP 1983/NA 14, Dept. of Applied Mathematics and Theoretical Physics, University of Cambridge (Cambridge, England, 1983). · Zbl 0531.65035
[8] D.F. Shanno, ”Conjugate gradient methods with inexact searches”,Mathematics of Operations Research 3 (1978) 244–256. · Zbl 0399.90077
[9] D.F. Shanno, ”On the convergence of a new conjugate gradient algorithm”,SIAM Journal of Numerical Analysis 15 (1978) 1247–1257. · Zbl 0408.90071
[10] D.F. Shanno and K.H. Phua, ”Remark on algorithm 500”,Transactions on Mathematical Software 6 (1980) 618–622.
[11] G. Zoutendijk, ”Nonlinear programming, computational methods”, in: J. Abadie, ed.,Integer and nonlinear programming (North-Holland, Amsterdam, 1970) pp. 37–86. · Zbl 0336.90057
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