We propose a new interior-point-based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems such as max–cut. Other applications include max–min eigenvalue problems and relaxations for the stable set problem.

A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically, semidefinite relaxations arising in combinatorial applications have sparse and well-structured cost and coefficient matrices of huge order. We present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time. Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completeness. We present numerical examples demonstrating the efficiency of the approach on combinatorial examples.