On equivalence of major relaxation methods for minimum ellipsoid covering intersection of ellipsoids

This paper investigates the problem on the minimum trace or volume ellipsoid covering intersection of ellipsoids. This problem can be solved by three major relaxation methods involving semi-definite programming relaxation, S-procedure relaxation and parameterized bounding ellipsoid relaxation. They are proposed from the different ideas or viewpoints, so that it is difficult to judge which relaxation method is the most suitable for the different practical implementations. To the best of our knowledge, the problem of proving the convexity of the optimization problem derived by the third relaxation method remains open. By rigorous analysis, we reveal the equivalence among the three relaxation methods, and address the open problem, i.e., the optimization problem derived by the third relaxation method can be equivalently transformed to a convex optimization problem. The computation complexity of these relaxation methods are also discussed, and the equivalent result is helpful to set membership filter and multisensor fusion obtaining a tighter ellipsoid to contain the state with lower computation complexity. Finally, the performance of each method is evaluated by several typical numerical examples in information fusion and filtering.