On the long-step path-following method for semidefinite programming

It has been shown in various recent research reports that the analysis of short-step primal–dual path-following algorithms for linear programming can be nicely generalized to semidefinite programming. However, the analysis of long-step path-following algorithms for semidefinite programming appeared to be less straightforward. For such an algorithm, Monteiro obtained an O(n^1.5 log(1/ε)) iteration bound for obtaining an ε-optimal solution, where n is the order of the semidefinite decision variable. In this paper, we propose to use a different search direction, viz. the so-called V-space direction. It is shown that this modification reduces the iteration complexity to O(n log(1/ε)). Independently, Monteiro and Y. Zhang obtained a similar result using Nesterov–Todd directions.