This paper extends an alternating projection algorithm for finding feasible points to block-diagonal semidefinite constraints consisting of linear equality constraints of block-diagonal matrices and a block-diagonal positive semidefinite matrix variable. This work is based on the 2007 work of Rami et al. on the semidefinite feasibility problem. We exploit problem structure and use eigenvalue shifting instead of eigenvalue replacement in the projection of symmetric matrices onto the cone of positive semidefinite matrices. We give numerical results that show the method is effective and works better with eigenvalue shifting than eigenvalue replacement. We also use numerical experiments to investigate the effect of a conification procedure on the method that was introduced in Rami et al. when finding strictly feasible points. The results indicate that the procedure might not be useful in practice, even though it guarantees convergence of MAP after a finite number of iterations.

linear matrix inequalities, feasibility, method of alternating projections, semidefinite programming.