Structured Primal-dual Interior-point Methods for Banded Semidefinite Programming

For semidefinite programming (SDP) problems, traditional primaldual interior-point methods based on conventional matrix operations have an upper limit on the problem size that the computer can handle due to memory constraints. But for a special kind of SDP problem, which is called the banded symmetric semidefinite programming (BSDP) problem, a memory-efficient algorithm, called a structured primal-dual interior-point method, can be applied. The method is based on the observation that both banded matrices and their inverses can be represented in sequentially semi-separable (SSS) form with numerical ranks equal to the half bandwidths of the banded matrices. Moreover, all computation can be done sequentially using the SSS form. Experiments of various problem sizes are performed to verify the feasibility of the proposed method.