Semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata

In this paper, we study the semidefinite inverse eigenvalue problem of reconstructing a real nn-by-nn matrix CC such that it is nearest to the original pre-estimated real nn-by-nn matrix CoCo in the Frobenius norm and satisfies the measured partial eigendata, where the required matrix CC should preserve the symmetry, positive semidefiniteness, and the prescribed entries of the pre-estimated matrix CoCo. 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 eigenvalue problems.