Semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata

In this paper, we study the semidefinite inverse eigenvalue problem of reconstructing a real n-by-n matrix C such that it is nearest to the original pre-estimated real n-by-n matrix Co in the Frobenius norm and satisfies the measured partial eigendata, where the required matrix C should preserve the symmetry, positive semidefiniteness, and the prescribed entries of the preestimated matrix Co. We propose the alternating direction method of multipliers for solving the semidefinite inverse eigenvalue problem, where three related iterative algorithms are presented. We also extend our method to the case of lower bounds. Numerical experiments are reported to illustrate the efficiency of the proposed method for solving semidefinite inverse problems.