Exploiting sparsity has been a key issue in solving large-scale optimization problems. The most time-consuming part of primal-dual interior-point methods for linear programs, second-order cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of the coefficient matrix of the equation that is determined by the sparsity of an optimization problem (linear program, semidefinite program or second-order program). We show if an optimization problem is correlatively sparse, then the coefficient matrix of the Schur complement equation inherits the sparsity, and a sparse Cholesky factorization applied to the matrix results in no fill-in.
Research Report B-434, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-29, Oh-Okayama, Meguro-ku, Tokyo 152-8552 Japan