Summary
It is shown that the theory developed in part I of this paper [22] can be applied to some well-known minimization algorithms with the quadratic termination property to prove theirn-step quadratic convergence. In particular, some conjugate gradient methods, the rank-1-methods of Pearson and McCormick (see Pearson [18]) and the large class of rank-2-methods described by Oren and Luenberger [16, 17] are investigated.
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Baptist, P.: Über das Konvergenzverhalten gewisser Update-Verfahren zur minimierung einer reelwertigen Funktion bei nichtperfekter Schrittlängenbestimmung. Diss., Universität Würzburg, 1975
Broyden, C. G.: Quasi-Newton methods and their application to function minimization. Math. Comput.21, 368–381 (1967)
Broyden, C. G.: The convergence of a class of double-rank minimization algorithms. Part 1 and Part 2. J. Inst. Math. Appl.6, 76–90, 222–231 (1970)
Burmeister, W.: Die Konvergenzordnung des Fletcher-Powell Algorithmus. Z. Angew. Math. Mech.53, 696–699 (1973)
Cohen, A.: Rate of convergence of several conjugate gradient algorithms. SIAM J. numer. Analysis9, 248–259 (1972)
Daniel, J.W.: The conjugate gradient method for linear and nonlinear operator equations. SIAM J. Numer. Anal.4, 10–26 (1967)
Davidon, W. C.: Variable metric method for minimization. Argonne Nat. Labs., Report ANL-5990, Rev. 1959
Dixon, L. C. W.: On quadratic termination and second order convergence: Two properties of unconstrained optimization algorithms. 1975
Fletcher, R., Powell, M. J. D.: A rapidly convergent descent method for minimization. Computer J.6, 163–168 (1963)
Fletcher, R., Reeves, C. M.: Function minimization by conjugate gradients. Comput. J.7, 145–154 (1964)
Lenard, M. L.: Practical convergence conditions for unconstrained optimization. Math. Programming4, 309–323 (1973)
Lenard, M. L.: Practical convergence conditions for the Davidon-Fletcher-Powell method. Math. Programming9, 69–86 (1975)
Lenard, M. L.: Convergence conditions for restarted conjugate gradient methods with inaccurate line searches. Math. Programming10, 32–51 (1976)
McCormick, G. P., Ritter, K.: Methods of conjugate directions versus quasi-Newton methods. Math. Programming3, 101–116 (1972)
McCormick, G. P., Ritter, K.: Alternative proofs of the convergence properties of the conjugategradient method. J. Optimization Theory Appl.13, 497–518 (1974)
Oren, S. S., Luenberger, D. G.: Self-scaling variable metric algorithms. Part I. Criteria and sufficient conditions for scaling a class of algorithms. Management Sci.20, 845–862 (1974)
Oren, S. S.: Self-scaling variable metric algorithms, Part II. Implementation and experiments. Management Sci.20, 863–874 (1974)
Pearson, J. D.: Variable metric methods of minimization. Comput. J.12, 171–178 (1969)
Polak, E., Ribière, G.: Note sur la convergence de methodes de directions conjugées. Rev. Francaise Informat. Recherche Operationelle (16-R1), pp. 35–43, 1969
Schuller, G., Stoer, J.: Über die Konvergenzordnung gewisser Rang-2-Verfahren zur Minimierung von Funktionen. In: International series of numerical mathematics, Vol.23, pp. 125–147. Basel: Birkhäuser 1974
Stoer, J.: On the convergence rate of imperfect minimization algorithms in Broyden's β-class. Math. Programming9, 313–335 (1975)
Stoer, J.: On the relation between quadratic termination and convergence properties of minimization algorithms. Part I. Theory. Numer. Math.28, 343–366 (1977)
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This work was supported in part at Stanford University, Stanford, California, under Energy Research and Development Administration, Contract E(04-3) 326 PA No. 30, and National Science Foundation Grant DCR 71-01996 A04 and in part by the Deutsche Forschungsgemeinschaft
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Baptist, P., Stoer, J. On the relation between quadratic termination and convergence properties of minimization algorithms. Numer. Math. 28, 367–391 (1977). https://doi.org/10.1007/BF01404342
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DOI: https://doi.org/10.1007/BF01404342