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Dualization

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Part of the Grundlehren der mathematischen Wissenschaften book series (GL,volume 317)

In the realm of convexity, almost every mathematical object can be paired with another, said to be dual to it. The pairing between convex cones and their polars has already been fundamental in the variational geometry of Chapter 6 in relating tangent vectors to normal vectors. The pairing between convex sets and sublinear functions in Chapter 8 has served as the vehicle for expressing connections between subgradients and subderivatives. Both correspondences are rooted in a deeper principle of duality for ‘conjugate’ pairs of convex functions, which will emerge fully here.

On the basis of this duality, close connections between otherwise disparate properties are revealed. It will be seen for instance that the level boundedness of one function in a conjugate pair corresponds to the finiteness of the other function around the origin. A catalog of such surprising linkages can be put together, and lists of dual operations and constructions to go with them.

Keywords

  • Convex Function
  • Dual Problem
  • Convex Cone
  • Support Function
  • Piecewise Linear

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1998 Springer-Verlag Berlin Heidelberg

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(1998). Dualization. In: Variational Analysis. Grundlehren der mathematischen Wissenschaften, vol 317. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02431-3_11

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  • DOI: https://doi.org/10.1007/978-3-642-02431-3_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62772-2

  • Online ISBN: 978-3-642-02431-3

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