Geodesic
Symmetric-rank-one multi-step quasi-Newton implicit update algorithms
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 3, pp. 241-248.

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Implicit multi-step quasi-Newton methods, introduced in [1], use the existing Hessian approximation to compute, at each iteration, the parameters required in the interpolation. To avoid the burden of computing the needed matrix-vector products, required by this approach, approximations based on the Secant Equation were proposed. Based on [2], a different approach to dealing with this difficulty was suggested, in which standard single-step quasi-Newton updates were replaced by successive iterations, by two-step updates, so that approximations were no longer necessary. The recent research has shown that the quantities required to compute the parameters referred to the above may be exactly computed by means of recurrence, so that the technique of alternation is no longer the only alternative. In this paper, we consider the derivation of new recurrences for the implicit update methods based on the well-known Symmetric Rank One (SRI) update formula. We present the results of a range of numerical experiments to compare and evaluate the methods developed here. 
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I. A Moughrabi. Symmetric-rank-one multi-step quasi-Newton implicit update algorithms. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 3, pp. 241-248. http://geodesic.mathdoc.fr/item/SJVM_2004_7_3_a5/

[1] Ford J. A., “Implicit updates in multi-step quasi-Newton methods”, Comput. Math. Appl., 42 (2001), 1083–1091 | DOI | MR | Zbl

[2] Ford J. A., Moghrabi I. A., “Alternating multi-step quasi-Newton methods for unconstrained optimization”, J. Comput. Appl. Math., 82 (1997), 105–116 | DOI | MR | Zbl

[3] Ford J. A., Moghrabi I. A., “Alternative parameter choices for multi-step quasi-Newton methods”, Optimization Methods and Software, 2 (1993), 357–370 | DOI | MR

[4] Broyden C. G., “The convergence of a class of double-rank minimization algorithms. Part 2: The new algorithm”, J. Inst. Math. Appl., 6 (1970), 222–231 | DOI | MR | Zbl

[5] Fletcher R., “A new approach to variable metric algorithms”, Comput. J., 13 (1970), 317–322 | DOI | Zbl

[6] Goldfarb G., “A family of variable metric methods derived by variational means”, Math. Comp., 24 (1970), 23–26 | DOI | MR | Zbl

[7] Shanno D. F., “Conditioning of quasi-Newton methods for function minimization”, Math. Comp., 24 (1970), 647–656 | DOI | MR

[8] Fletcher R., Practical Methods of Optimization, 2nd ed., Wiley, New York, 1987 | MR

[9] Shanno D. F., Phua K. H., “Matrix conditioning and nonlinear optimization”, Math. Prog., 14 (1978), 149–160 | DOI | MR | Zbl

[10] Ford J. A., Moghrabi I. A., “Multi-step quasi-Newton methods for optimization”, J. Comput. Appl. Math., 50 (1994), 305–323 | DOI | MR | Zbl

[11] Ford J. A., Moghrabi L. A., “Minimum curvature multi-step quasi-Newton methods”, Comput. Math. Appl., 31 (1996), 179–186 | DOI | MR | Zbl

[12] Dennis J. E., Schnabel R. B., “Least change secant updates for quasi-Newton methods”, SIAM Review, 21 (1979), 443–459 | DOI | MR | Zbl

[13] More J. J., Garbow B. S., Hillstrom K. E., “Testing unconstrained optimization software”, ACM Trans. Math. Software, 7 (1981), 17–41 | DOI | MR | Zbl

[14] Conn A. R., Gould N. I. M., Toint Ph. L., Testing a class of methods for solving minimization problems with simple bounds on the variables, Research Report CS-86-45, University of Waterloo, 1986

[15] Toint Ph. L., “On large scale nonlinear least squares calculations”, SIAM J. Sci. Stat. Comput., 8 (1987), 416–435 | DOI | MR | Zbl