Conditions for the existence, identification and calculus rules of the threshold of prox-boundedness

Abstract

This note advances knowledge of the threshold of prox-boundedness of a function; an important concern in the use of proximal point optimization algorithms and in determining the existence of the Moreau envelope of the function. In finite dimensions, we study general prox-bounded functions and then focus on some useful classes such as piecewise functions and Lipschitz continuous functions. The thresholds are explicitly determined when possible and bounds are established otherwise. Some calculus rules are constructed; we consider functions with known thresholds and find the thresholds of their sum and composition.

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Acknowledgements

The author wishes to thank the anonymous referees for their valued input in the improvement of this work and to Referee 1 in particular for suggesting a cleaner, more concise proof of Proposition 3.2.

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Correspondence to C. Planiden.

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Planiden, C. Conditions for the existence, identification and calculus rules of the threshold of prox-boundedness. Optim Lett 15, 45–57 (2021). https://doi.org/10.1007/s11590-020-01583-2

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Keywords

  • Fenchel conjugate
  • Infimal convolution
  • Lipschitz continuous
  • Moreau envelope
  • Moreau–Yosida regularization
  • Piecewise function
  • Prox-bounded
  • Proximal mapping
  • Regularization
  • Threshold