Solving SDP's in Non-commutative Algebras Part I : The Dual-Scaling Algorithm

Semidefinite programming (SDP) may be viewed as an extension of linear programming (LP), and most interior point methods (IPM s) for LP can be extended to solve SDP problems.However, it is far more difficult to exploit data structures (especially sparsity) in the SDP case.In this paper we will look at the data structure where the SDP data matrices lie in a low dimensional matrix algebra.This data structure occurs in several applications, including the lower bounding of the stability number in certain graphs and the crossing number in complete bipartite graphs.We will show that one can reduce the linear algebra involved in an iteration of an IPM to involve matrices of the size of the dimension of the matrix algebra only.In other words, the original sizes of the data matrices do not appear in the computational complexity bound.In particular, we will work out the details for the dual scaling algorithm, since a dual method is most suitable for the types of applications we have in mind.