Abstract
In order to apply quasi-Newton methods to solve unconstrained minimization calculations when the number of variables is very large, it is usually necessary to make use of any sparsity in the second derivative matrix of the objective function. Therefore, it is important to extend to the sparse case the updating formulae that occur in variable metric algorithms to revise the estimate of the second derivative matrix. Suitable extensions suggest themselves when the updating formulae are derived by variational methods [1, 3]. The purpose of the present paper is to give a new proof of a theorem of Dennis and Schnabel [1], that shows the effect of sparsity on updating formulae for second derivative estimates.
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References
J.E. Dennis and R.B. Schnabel, “Least change secant updates for quasi—Newton methods”,SIAM Review 21 (1979) 443–459.
J. Greenstadt, “Variations on variable metric methods”,Mathematics of Computation 24 (1970) 1–22.
Ph.L. Toint, “On sparse and symmetric matrix updating subject to a linear equation”,Mathematics of Computation 31 (1977) 954–961.
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Powell, M.J.D. A note on quasi-newton formulae for sparse second derivative matrices. Mathematical Programming 20, 144–151 (1981). https://doi.org/10.1007/BF01589341
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DOI: https://doi.org/10.1007/BF01589341