Symmetric primal-dual path following algorithms for semidefinite programming

Jos Sturm and Shuzhong Zhang

In this paper a symmetric primal-dual transformation for positive semidefinite programming is proposed. For standard SDP problems, after this symmetric transformation the primal variables and the dual slacks become identical. In the context of linear programming, existence of such a primal-dual transformation is a well known fact. Based on this symmetric primal-dual transformation we derive Newton search directions for primal-dual path-following algorithms for semidefinite programming. In particular, we generalize: (1) the short step path following algorithm, (2) the predictor-corrector algorithm and (3) the largest step algorithm to semidefinite programming. It is shown that these algorithms require at most ${\cal O}(\sqrt{n}\mid \log \epsilon \mid )$ main iterations for computing an $\epsilon $-optimal solution. The symmetric primal-dual transformation discussed in this paper can be interpreted as a specialization of the scaling-point concept introduced by Nesterov and Todd \cite{NT95} for self-scaled conic problems. The difference is that we explicitly use the usual $v$-space notion and the proofs look very similar to the linear programming case.

Report 9554/A, Econometric Institute, Erasmus University, Rotterdam.