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Apart Sets and Functions: an Application to the Stability of Penalized Optimization Problems

Abstract

The notion of apart sets and functions, a technical concept introduced by Luc and Penot, is studied in the framework of star-shaped sets and functions. As an application, we provide a necessary and sufficient condition ensuring that a penalized optimization problem is stable with respect to perturbations of both the penalty and the objective functions, in the case when the perturbations are small in the sense of the Attouch–Wets topology.

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Acknowledgements

We would like to warmly thank the anonymous referee. His careful reading of the paper allowed us to correct a significant number of typos and errors and largely contributed to the final form of the proofs of several important results of the article.

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Correspondence to Emil Ernst.

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Dedicated to Michel Théra in honor of his 70th birthday.

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Ernst, E. Apart Sets and Functions: an Application to the Stability of Penalized Optimization Problems. Vietnam J. Math. 46, 343–358 (2018). https://doi.org/10.1007/s10013-018-0288-9

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  • DOI: https://doi.org/10.1007/s10013-018-0288-9

Keywords

  • Apart sets and functions
  • Star-shaped sets and functions
  • Constrained optimization
  • Penalty function
  • Stability
  • Attouch–Wets topology

Mathematics Subject Classification (2010)

  • 49M30
  • 49N15
  • 52A20