Some results on semidefinite programming with rank constraint

Rank constraints often make an optimization problem hard in generally and very possibly NP-hard. In this paper, suppose that A₁,..., A_m are symmetric positive semidefinite n × n matrices with positive definite sum and A is a positive semidefinite symmetric n × n matrix, then, we have proved that the problem, max over x {Tr(AX)|Tr(A_iX) ≤ 1, i = 1, 2, ..., m; X ≥ 0; rank (X) ≤ min {r,m}}, can be solved in polynomial time, where r denotes the rank of A.