Symmetric primal - dual path following

In this paper a symmetric primal-dual transformation for positive semideenite 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 semideenite programming. In particular, we generalize: (1) the short step path following algorithm, (2) the predictor-corrector algorithm and (3) the largest step algorithm to semideenite programming. It is shown that these algorithms require at most O(p n j log j) main iterations for computing an-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 12] for self-scaled conic problems. The diierence is that we explicitly use the usual v-space notion and the proofs look very similar to the linear programming case.