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Variational Analysis on the Signed Distance Functions

Author

Listed:
  • Honglin Luo

    (Chongqing Normal University)

  • Xianfu Wang

    (University of British Columbia
    Southwest University)

  • Brett Lukens

    (University of British Columbia)

Abstract

The signed distance function (or oriented distance function) of a set in a metric space determines the distance of a given point from the boundary of the set, with the sign determined by whether the point is in the set or in its complement. The knowledge of signed distance functions is a very valuable information in various fields of applied mathematics such as collision detection, binary classification, shape analysis, fuzzy numbers ranking and level set methods. One distinguished feature of the signed distance function is that it reflects the geometric structure of the set much better than the distance function does. We explore many interesting analytical properties of signed distance functions and use them to construct convex functions with not convex subdifferential domains. Several examples are presented to illustrate most of these fine properties.

Suggested Citation

  • Honglin Luo & Xianfu Wang & Brett Lukens, 2019. "Variational Analysis on the Signed Distance Functions," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 751-774, March.
    • Handle: RePEc:spr:joptap:v:180:y:2019:i:3:d:10.1007_s10957-018-1414-2
    DOI: 10.1007/s10957-018-1414-2
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    References listed on IDEAS

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    1. Lkhamsuren Altangerel & Gert Wanka & Oleg Wilfer, 2013. "An Oriented Distance Function Application to Gap Functions for Vector Variational Inequalities," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Rentsen Enkhbat & E. N. Pistikopoulos (ed.), Optimization, Simulation, and Control, edition 127, pages 17-34, Springer.
    2. J. B. Hiriart-Urruty, 1979. "Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 4(1), pages 79-97, February.
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    Cited by:

    1. Qamrul Hasan Ansari & Pradeep Kumar Sharma, 2022. "Some Properties of Generalized Oriented Distance Function and their Applications to Set Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 247-279, June.

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