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On global minimizers of quadratic functions with cubic regularization

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Abstract

In this paper, we analyze some theoretical properties of the problem of minimizing a quadratic function with a cubic regularization term, arising in many methods for unconstrained and constrained optimization that have been proposed in the last years. First we show that, given any stationary point that is not a global solution, it is possible to compute, in closed form, a new point with a smaller objective function value. Then, we prove that a global minimizer can be obtained by computing a finite number of stationary points. Finally, we extend these results to the case where stationary conditions are approximately satisfied, discussing some possible algorithmic applications.

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Authors and Affiliations

  1. Department of Mathematics, University of Padua, Via Trieste, 63, 35121, Padua, Italy

    Andrea Cristofari

  2. Department of Mathematics, Yazd University, P.O. Box 89195-74, Yazd, Iran

    Tayebeh Dehghan Niri

  3. Department of Computer, Control and Management Engineering, Sapienza University of Rome, Via Ariosto, 25, 00185, Rome, Italy

    Stefano Lucidi

Authors
  1. Andrea Cristofari
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  2. Tayebeh Dehghan Niri
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  3. Stefano Lucidi
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Correspondence to Andrea Cristofari.

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Cristofari, A., Dehghan Niri, T. & Lucidi, S. On global minimizers of quadratic functions with cubic regularization. Optim Lett 13, 1269–1283 (2019). https://doi.org/10.1007/s11590-018-1316-0

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  • DOI: https://doi.org/10.1007/s11590-018-1316-0

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