  This paper investigates the problem on the minimum ellipsoid containing the
intersection of multiple ellipsoids, which has been extensively applied to
information science, target tracking and data fusion etc. There are three major
relaxation methods involving SDP relaxation, S-procedure relaxation and
bounding ellipsoid relaxation, which are derived by different ideas or
viewpoints. However, it is unclear for the interrelationships among these
methods. This paper reveals the equivalence among the three relaxation methods
by three stages. Firstly, the SDP relaxation method can be equivalently
simplified to a decoupled SDP relaxation method. Secondly, the equivalence
between the SDP relaxation method and the S-procedure relaxation method can be
obtained by rigorous analysis. Thirdly, we establish the equivalence between
the decoupled SDP relaxation method and the bounding ellipsoid relaxation
method. Therefore, the three relaxation methods are unified through the
decoupled SDP relaxation method. By analysis of the computational complexity,
the decoupled SDP relaxation method has the least computational burden among
the three methods. The above results are helpful for the research of
set-membership filter and distributed estimation fusion. Finally, the
performance of each method is evaluated by some typical numerical examples in
information fusion and filtering.
