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On the convergence rate of imperfect minimization algorithms in Broyden’s \(\beta\)-class. (English) Zbl 0346.90047

MSC:
90C30 Nonlinear programming
41A15 Spline approximation
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[1] C.G. Broyden, ”Quasi-Newton methods and their application to function minimization”,Mathematics of Computation 21 (1967) 368–381. · Zbl 0155.46704 · doi:10.1090/S0025-5718-1967-0224273-2
[2] C.G. Broyden, ”The convergence of a class of double-rank minimization algorithms”, Parts 1 and 2,Journal of the Institute of Mathematics and its Applications 6 (1970) 76–90, 222–231. · Zbl 0223.65023 · doi:10.1093/imamat/6.1.76
[3] C.G. Broyden, J.E. Dennis, Jr. and J.J. Mord, ”On the local and superlinear convergence of Quasi-Newton methods”,Journal of the Institute of Mathematics and its Applications 12 (1973) 223–245. · Zbl 0282.65041 · doi:10.1093/imamat/12.3.223
[4] W. Burmeister, ”Die Konvergenzordnung des Fletcher-Powell Algorithmus”,Zeitschrift für Angewandte Mathematik und Mechanik 53 (1973) 696–699. · Zbl 0269.90039
[5] A. Cohen, ”Rate of convergence of several conjugate gradient algorithms”,SIAM Journal on Numerical Analysis 9 (1972) 248–259. · Zbl 0279.65051 · doi:10.1137/0709024
[6] W.C. Davidon, ”Variable metric method for minimization”, Argonne National Laboratories rept. ANL-5990 (1959). · Zbl 0752.90062
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[8] J.E. Dennis, Jr., ”On some methods based on Broyden’s secant approximation to the Hessian”, in: F.A. Lootsma, ed.,Numerical methods for non-linear optimization (Academic Press, London, 1972).
[9] J.E. Dennis, Jr. and J.J. More, ”A characterization of superlinear convergence and its applications to Quasi-Newton methods”,Mathematics of Computation 28 (1974) 549–560. · Zbl 0282.65042 · doi:10.1090/S0025-5718-1974-0343581-1
[10] L.C.W. Dixon, ”Variable metric algorithms: necessary and sufficient conditions for identical behaviour on non-quadratic functions”,Journal of Optimization Theory and Applications 10 (1972) 34–40. · Zbl 0226.49014 · doi:10.1007/BF00934961
[11] R. Fletcher and M.J.D. Powell, ”A rapidly convergent descent method for minimization”,The Computer Journal 6 (1963) 163–168. · Zbl 0132.11603
[12] R. Fletcher, ”A new approach to variable metric algorithms”,The Computer Journal 13 (1970) 317–322. · Zbl 0207.17402 · doi:10.1093/comjnl/13.3.317
[13] H.Y. Huang, ”Unified approach to quadratically convergent algorithms for function minimization”,Journal of Optimization Theory and Applications 5 (1970) 405–423. · Zbl 0194.19402 · doi:10.1007/BF00927440
[14] J.M. Ortega and W.C. Rheinboldt,Iterative solution of non-linear equations in several variables (Academic Press, New York, 1970). · Zbl 0241.65046
[15] M.J.D. Powell, ”On the convergence of the variable metric algorithm”,Journal of the Institute of Mathematics and its Applications 7 (1971) 21–36. · Zbl 0217.52804 · doi:10.1093/imamat/7.1.21
[16] M.J.D. Powell, ”Some properties of the variable metric algorithm”, in: F.A. Lootsma, ed.,Numerical methods for non-linear optimization (Academic Press, London, 1972). · Zbl 0288.65036
[17] W.C. Rheinboldt and J.S. Vandergraft, ”On the local convergence of update methods”, Tech. rept. TR 225, Computer Science Center, University of Maryland, College Park, Md. (1973). · Zbl 0255.15017
[18] G. Schuller, ”On the order of convergence of certain Quasi-Newton methods”,Numerische Mathematik 23 (1974) 181–192. · Zbl 0292.65034 · doi:10.1007/BF01459951
[19] G. Schuller and J. Steer,Über die Konvergenzordnung gewisser Rang-2 Verfahren zur Minimierung von Funktionen, International Series of Numerical Mathematics, Vol. 23 (Birkhäuser, Basel, 1974) pp. 125–147. · Zbl 0331.65042
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