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A Newton-like method with mixed factorizations and cubic regularization for unconstrained minimization

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Abstract

A Newton-like method for unconstrained minimization is introduced in the present work. While the computer work per iteration of the best-known implementations may need several factorizations or may use rather expensive matrix decompositions, the proposed method uses a single cheap factorization per iteration. Convergence and complexity and results, even in the case in which the subproblems’ Hessians are far from being Hessians of the objective function, are presented. Moreover, when the Hessian is Lipschitz-continuous, the proposed method enjoys \(O(\varepsilon ^{-3/2})\) evaluation complexity for first-order optimality and \(O(\varepsilon ^{-3})\) for second-order optimality as other recently introduced Newton method for unconstrained optimization based on cubic regularization or special trust-region procedures. Fairly successful and fully reproducible numerical experiments are presented and the developed corresponding software is freely available.

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Notes

  1. With the exception of problem ARGLINC for which we set \(f_{\mathrm {target}}\) as its known optimal value times \(1+10^{-15}\).

  2. With the exception of problem STREC that we were unable to find in the current distribution of CUTEst.

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Acknowledgements

The authors would like to thank H. D. Scolnik for providing an updated version of the method introduced in [15]. We also would like to thank N. I. M. Gould for pointing out the work [21] and also for pointing out the usage of subroutine MA57_get_factors in the sparse implementation of the BPK-based Mixed Factorization. Finally, the authors would like to thank the reviewers for their helpful comments.

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Authors and Affiliations

  1. Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, Cidade Universitária, São Paulo, SP, 05508-090, Brazil

    E. G. Birgin

  2. Department of Applied Mathematics, Institute of Mathematics, Statistics, and Scientific Computing, State University of Campinas, Campinas, SP, 13083-859, Brazil

    J. M. Martínez

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This work was supported by FAPESP (Grants 2013/05475-7, 2013/07375-0, 2016/01860-1, and 2018/24293-0) and CNPq (Grants 309517/2014-1 and 303750/2014-6).

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Birgin, E.G., Martínez, J.M. A Newton-like method with mixed factorizations and cubic regularization for unconstrained minimization. Comput Optim Appl 73, 707–753 (2019). https://doi.org/10.1007/s10589-019-00089-7

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