Incorporating Inequality Constraints in the Spectral Bundle Method

Semidefinite relaxations of quadratic 0-1 programming or graph partitioning problems are well known to be of high quality. However, solving them by primal-dual interior point methods can take much time even for problems of moderate size. Recently we proposed a spectral bundle method that allows to compute, within reasonable time, approximate solutions to structured, large equality constrained semidefinite programs if the trace of the primal matrix variable is fixed. The latter property holds for the aforementioned applications. We extend the spectral bundle method so that it can handle inequality constraints without seriously increasing computation time. This makes it possible to apply cutting plane algorithms to semidefinite relaxations of real world sized instances. We illustrate the efficacy of the approach by giving some preliminary computational results.