## Nonsymmetric Search Directions for Semidefinite Programming

### Rongqin Sheng and Florian A. Potra

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.

Contact: sheng@mcs.anl.gov