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

TL;DR

This paper demonstrates the equivalence of three major relaxation methods for finding the minimum ellipsoid containing the intersection of multiple ellipsoids, unifying their theory and analyzing their computational efficiency.

Contribution

It reveals the equivalence among SDP, S-procedure, and bounding ellipsoid relaxation methods and identifies the decoupled SDP method as the most computationally efficient.

Findings

All three relaxation methods are equivalent through a unified framework.

Decoupled SDP relaxation has the lowest computational complexity.

Numerical examples confirm the theoretical results and practical efficiency.

Abstract

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.