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This paper investigates the problem on the minimum ellipsoid containing theintersection of multiple ellipsoids, which has been extensively applied toinformation science, target tracking and data fusion etc. There are three majorrelaxation methods involving SDP relaxation, S-procedure relaxation andbounding ellipsoid relaxation, which are derived by different ideas orviewpoints. However, it is unclear for the interrelationships among thesemethods. This paper reveals the equivalence among the three relaxation methodsby three stages. Firstly, the SDP relaxation method can be equivalentlysimplified to a decoupled SDP relaxation method. Secondly, the equivalencebetween the SDP relaxation method and the S-procedure relaxation method can beobtained by rigorous analysis. Thirdly, we establish the equivalence betweenthe decoupled SDP relaxation method and the bounding ellipsoid relaxationmethod. Therefore, the three relaxation methods are unified through thedecoupled SDP relaxation method. By analysis of the computational complexity,the decoupled SDP relaxation method has the least computational burden amongthe three methods. The above results are helpful for the research ofset-membership filter and distributed estimation fusion. Finally, theperformance of each method is evaluated by some typical numerical examples ininformation fusion and filtering.

On Equivalence and Computational Efficiency of the Major Relaxation Methods for Minimum Ellipsoid Containing the Intersection of Ellipsoids