IPRSDP: a primal-dual interior-point relaxation algorithm for semidefinite programming

We propose an efficient primal-dual interior-point relaxation algorithm based on a smoothing barrier augmented Lagrangian, called IPRSDP, for solving semidefinite programming problems in this paper. The IPRSDP algorithm has three advantages over classical interior-point methods. Firstly, IPRSDP does not require the iterative points to be positive definite. Consequently, it can easily be combined with the warm-start technique used for solving many combinatorial optimization problems, which require the solutions of a series of semidefinite programming problems. Secondly, the search direction of IPRSDP is symmetric in itself, and hence the symmetrization procedure is not required any more. Thirdly, with the introduction of the smoothing barrier augmented Lagrangian function, IPRSDP can provide the explicit form of the Schur complement matrix. This enables the complexity of forming this matrix in IPRSDP to be comparable to or lower than that of many existing search directions. The global convergence of IPRSDP is established under suitable assumptions. Numerical experiments are made on the SDPLIB set, which demonstrate the efficiency of IPRSDP.