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The Geometry of Semidefinite Programming

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Handbook of Semidefinite Programming

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 27))

Abstract

Consider the primal-dual pair of optimization problems

$$ \begin{gathered} Min \left\langle {c,x} \right\rangle {\rm M}ax \left\langle {b,y} \right\rangle \hfill \\ (P) s.t. x \in K s.t. z \in K* (D) \hfill \\ Ax = b A*y + z = c \hfill \\ \end{gathered} $$

where

  • X and Y are Euclidean spaces with dim X ≥ dim Y.

  • A : XY is a linear operator, assumed to be onto.

  • A* : YX is its adjoint.

  • K is a closed, convex, facially exposed cone in X.

  • K* := {z|〈z,x〉≤ 0 ∀xK} is the dual of K, also a closed, convex, facially exposed cone.

My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful. Hermann Weyl (1985 – 1955) quoted in an obituary by Freeman J. Dyson in Nature, March 10, 1956 [Inscription above Plato’s Academy:] Let no one ignorant of geometry enter here. Plato

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© 2000 Springer Science+Business Media New York

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Pataki, G. (2000). The Geometry of Semidefinite Programming. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds) Handbook of Semidefinite Programming. International Series in Operations Research & Management Science, vol 27. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4381-7_3

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  • DOI: https://doi.org/10.1007/978-1-4615-4381-7_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-6970-7

  • Online ISBN: 978-1-4615-4381-7

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