Journal of Global Optimization

, Volume 69, Issue 4, pp 889–909 | Cite as

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

  • D. Russell Luke
  • Shoham Sabach
  • Marc Teboulle
  • Kobi Zatlawey


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.


Nonsmooth nonconvex minimization Kurdyka–Łojasiewicz property Method of multipliers Alternating minimization Convergence in semialgebraic optimization Single source localization 

Mathematics Subject Classification

90C26 90C90 49M37 65K05 


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • D. Russell Luke
    • 1
  • Shoham Sabach
    • 2
  • Marc Teboulle
    • 3
  • Kobi Zatlawey
    • 3
  1. 1.Institut für Numerische und Angewandte MathematikUniversität GöttingenGöttingenGermany
  2. 2.Faculty of Industrial Engineering and ManagementTechnion—Israel Institute of TechnologyHaifaIsrael
  3. 3.School of Mathematical SciencesTel-Aviv UniversityRamat-AvivIsrael

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