Nonsymmetric search directions for semidefinite programming

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, in many cases, requires less CPU time than the XZ+ZX method of Alizadeh, Overton, and Haeberly {[}SIAM J. Optim., 8 (1998), pp. 746-768] while achieving similar accuracy.