Extension of primal–dual interior point algorithms to symmetric cones

Extension of primal–dual interior point algorithms to symmetric cones

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Article ID: iaor20041760
Country: Germany
Volume: 96
Issue: 3
Start Page Number: 409
End Page Number: 438
Publication Date: Jan 2003
Journal: Mathematical Programming
Authors: ,
Keywords: interior point methods
Abstract:

In this paper we show that the so-called commutative class of primal–dual interior point algorithms which were designed by Monteiro and Zhang for semidefinite programming extends word-for-word to optimization problems over all symmetric cones. The machinery of Euclidean Jordan algebras is used to carry out this extension. Unlike some non-commutative algorithms such as the XS+SX method, this class of extensions does not use concepts outside of the Euclidean Jordan algebras. In particular no assumption is made about representability of the underlying Jordan algebra. As a special case, we prove polynomial iteration complexities for variants of the short-, semi-long-, and long-step path-following algorithms using the Nesterov–Todd XS, or SX directions.

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