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Extension of primal-dual interior point algorithms to symmetric cones. (English) Zbl 1023.90083
Summary: In this paper we show that the so-called commutative class of primal-dual interior point algorithms which were designed by R. D. C. Monteiro and Y. Zhang [Math. Program. 81A, 281-299 (1998; Zbl 0919.90109)] 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. See also the authors’ companion paper in Math. Oper. Res. 26, 543-564 (2001; Zbl 1073.90572).

##### MSC:
 90C51 Interior-point methods 90C22 Semidefinite programming 17C55 Finite-dimensional structures of Jordan algebras
##### Citations:
Zbl 0919.90109; Zbl 1073.90572
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