In a 2007 paper, Rami et al. gave a variant of the alternating projection algorithm for solving the semidefinite feasibility problem with a single linear matrix inequality constraint. We investigate the effect of the value used for eigenvalue replacement in the projection of symmetric matrices onto the cone of positive semidefinite matrices. We give numerical evidence indicating that the choices of 0 and 1 for eigenvalue replacement in the projection are not the best. A modification of the algorithm is also proposed where eigenvalue shift is used instead of eigenvalue replacement. We extend the algorithm to a system of linear matrix inequalities and give an implementation for the special case of second-order cone constraints. Finally, we present the results of numerical experiments which indicate that our modifications are effective.

linear matrix inequalities, feasibility, method of alternating projections, semidefinite programming, second-order cones.