Heuristic Bundle Upper Bound Based Polyhedral Bundle Method for Semidefinite Programming

Abstract: Semidefinite programming (SDP) is a fundamental class of convex optimization problems with diverse applications in mathematics, engineering, machine learning, and related disciplines. This paper investigates the application of the polyhedral bundle method to standard SDPs. The basic idea of this method is to approximate semidefinite constraints using linear constraints, and thereby transform the SDP problem into a series of quadratic programming subproblems. However, the number of linear constraints often increases continuously during the iteration process, leading to a significant reduction in the solution efficiency. Therefore, based on the idea of limiting the upper bound on the number of bundles, we heuristically select the upper bound through numerical experiments according to the rank of the primal optimal solution and propose a modified subproblem. In this way, under the premise of ensuring the approximation ability of the lower approximation model, we minimize the number of bundles as much as possible to improve the solution efficiency of the subproblems. The algorithm performs well in both Max-Cut problems and random SDP problems. In particular, for random sparse SDP problems with low condition numbers, under the condition of a relative accuracy of (10^{-4}), it shows a significant improvement compared with algorithms such as interior-point methods.