A fixed-point method for approximate projection onto the positive semidefinite cone

The projection of a symmetric matrix onto the positive semidefinite cone is an important problem with application in many different areas such as economy, physics and, directly, semidefinite programming. This problem has analytical solution, but it relies on the eigendecomposition of a given symmetric matrix which clearly becomes prohibitive for larger dimension and dense matrices. We present a fixed-point iterative method for computing an approximation of such projection. Each iteration requires matrix-matrix products whose costs may be much less than O(n3) for certain structured matrices. Numerical experiments showcase the attractiveness of the proposed approach for sparse symmetric banded matrices.