This paper is concerned with a class of ellipsoidal sets (ellipsoids and elliptic cylinders) in the m-dimensional Euclidean space which are determined by a freely chosen positive semidefinite matrix. All ellipsoidal sets in this class are similar to each other through a parallel transformation and a scaling around their centers by a constant factor. Based on the basic idea of lifting, we first present a conceptual min-max problem to determine an ellipsoidal set with the smallest size in this class which encloses a given subset of the m-dimensional Euclidean space. Then we derive a numerically tractable enclosing ellipsoidal set of a given semialgebraic set as a convex relaxation of the min-max problem in the lifting space. A main feature of the proposed method is that it is designed to incorporate into existing SDP relaxations with exploiting sparsity for various optimization problems to compute error bounds of their optimal solutions. We discuss how we adapt the method to a sparse variant of Lasserre's hierarchy SDP relaxation for polynomial optimization problems and to a standard SDP relaxation for quadratic optimization problems. Some numerical results on polynomial optimization problems and the sensor network localization problem are also presented.
Research Report B-459, Department of Mathematical and Computing Science, Tokyo Institute of Technology, Meguro, Tokyo 152-8552, Japan