Bipolar theorem

In mathematics, the bipolar theorem is a theorem in convex analysis which provides necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.[1]:76–77

Statement of theorem

For any nonempty set in some linear space , then the bipolar cone is given by

where denotes the convex hull.[1]:54[2]

Special case

is a nonempty closed convex cone if and only if when , where denotes the positive dual cone.[2][3]

Or more generally, if is a nonempty convex cone then the bipolar cone is given by

Relation to the Fenchel–Moreau theorem

If is the indicator function for a cone . Then the convex conjugate is the support function for , and . Therefore, if and only if .[1]:54[3]

References

  1. 1 2 3 Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701.
  2. 1 2 Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 9780521833783. Retrieved October 15, 2011.
  3. 1 2 Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.