Two nonsymmetric search directions for semidefinite programming, the XZ and ZX search directions, are proposed. They are derived from a nonsymmetric formulation of the semidefinite programming problem. The XZ direction corresponds to the direct linearization of the central path equation $XZ = \nu I,$ while the ZX direction corresponds to $ZX = \nu I$. The XZ and ZX directions are well defined if both $X$ and $Z$ are positive definite matrices, where $X$ may be nonsymmetric. We present an algorithm using the XZ and ZX directions alternately following the Mehrotra predictor-corrector framework. Numerical results show that the XZ/ZX algorithm is, in most cases, faster than the XZ+ZX method of Alizadeh, Overton, and Haeberly (AHO) while achieving similar accuracy.
Preprint ANL/MCS-P692-0997, Mathematics and Computer Science Division, Argonne National Laboratory, September 1997.