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A simple globally convergent algorithm for the nonsmooth nonconvex single source localization problem

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Abstract

We study the single source localization problem which consists of minimizing the squared sum of the errors, also known as the maximum likelihood formulation of the problem. The resulting optimization model is not only nonconvex but is also nonsmooth. We first derive a novel equivalent reformulation as a smooth constrained nonconvex minimization problem. The resulting reformulation allows for deriving a delightfully simple algorithm that does not rely on smoothing or convex relaxations. The proposed algorithm is proven to generate bounded iterates which globally converge to critical points of the original objective function of the source localization problem. Numerical examples are presented to demonstrate the performance of our algorithm.

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Notes

  1. By the subdifferential \(\partial \varphi \) we mean the limiting subdifferential, which is defined for proper and lower semicontinuous functions by

    $$\begin{aligned} \partial \varphi \left( {\bar{z}}\right) :=\left\{ v : \, \exists v^{k} \rightarrow v \text{ and } z^{k} \mathop {\rightarrow }\limits ^{\varphi } {\bar{z}} \text{ such } \text{ that } \varphi \left( z \right) \ge \varphi \left( z^{k}\right) + \left\langle {v^{k} , z - z^{k}} \right\rangle + o(|z - z^{k}|) \right\} . \end{aligned}$$

    When \(\varphi \) is convex, the above definition coincides with the subdifferential of convex analysis. For more details see [23].

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Authors and Affiliations

  1. Institut für Numerische und Angewandte Mathematik, Universität Göttingen, 37083, Göttingen, Germany

    D. Russell Luke

  2. Faculty of Industrial Engineering and Management, Technion—Israel Institute of Technology, 3200003, Haifa, Israel

    Shoham Sabach

  3. School of Mathematical Sciences, Tel-Aviv University, 69978, Ramat-Aviv, Israel

    Marc Teboulle & Kobi Zatlawey

Authors
  1. D. Russell Luke
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  2. Shoham Sabach
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  3. Marc Teboulle
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  4. Kobi Zatlawey
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Corresponding author

Correspondence to Shoham Sabach.

Additional information

Supported by a GIF Grant G-1253-304.6/2014. Partially supported by the Israel Science Foundation, ISF Grant 998/12.

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Luke, D.R., Sabach, S., Teboulle, M. et al. A simple globally convergent algorithm for the nonsmooth nonconvex single source localization problem. J Glob Optim 69, 889–909 (2017). https://doi.org/10.1007/s10898-017-0545-6

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  • DOI: https://doi.org/10.1007/s10898-017-0545-6

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