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Properties of the block BFGS update and its application to the limited-memory block BNS method for unconstrained minimization

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Abstract

A block version of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) variable metric update formula and its modifications are investigated. In spite of the fact that this formula satisfies the quasi-Newton conditions with all used difference vectors and that the improvement of convergence is the best one in some sense for quadratic objective functions, for general functions, it does not guarantee that the corresponding direction vectors are descent directions. To overcome this difficulty, but at the same time utilize the advantageous properties of the block BFGS update, a block version of the limited-memory variable metric BNS method for large-scale unconstrained optimization is proposed. The global convergence of the algorithm is established for convex sufficiently smooth functions. Numerical experiments demonstrate the efficiency of the new method.

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Acknowledgments

We thank the anonymous referee for careful reading of the paper and for constructive suggestions.

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Authors and Affiliations

  1. Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží 2, 182 07, Prague 8, Czech Republic

    Jan Vlček & Ladislav Lukšan

  2. Technical University of Liberec, Hálkova 6, 461 17, Liberec, Czech Republic

    Ladislav Lukšan

Authors
  1. Jan Vlček
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  2. Ladislav Lukšan
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Correspondence to Jan Vlček.

Additional information

This work was supported by the Grant Agency of the Czech Republic, project No. GA13-06684S, and the Institute of Computer Science of the CAS (RVO: 67985807).

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Vlček, J., Lukšan, L. Properties of the block BFGS update and its application to the limited-memory block BNS method for unconstrained minimization. Numer Algor 80, 957–987 (2019). https://doi.org/10.1007/s11075-018-0513-3

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  • DOI: https://doi.org/10.1007/s11075-018-0513-3

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