IPRSOCP: A Primal-Dual Interior-Point Relaxation Algorithm for Second-Order Cone Programming

Inspired by the smoothing barrier augmented Lagrangian function in Liu et al. (Math Methods Oper Res 96(3):351–382, 2022), we propose a primal-dual interior-point relaxation algorithm for second-order cone programming, called IPRSOCP. Two features of the IPRSOCP algorithm are as follows. One is that the iterative points of the proposed algorithm need not lie inside the interior region, convening the use of warm-start. The other is that an explicit form of the Schur complement matrix is explored such that the low-rank structure of the Schur complement matrix can be used to improve numerical stability and efficiency. Under suitable assumptions, it is shown that the barrier parameter in the IPRSOCP algorithm tends to zero and the generated sequence of iterations is globally convergent to the solution. Numerical results demonstrate that the IPRSOCP algorithm is competitive with the open-source benchmark solvers, SeDuMi, SDPT3, and ECOS, in terms of robustness and efficiency.