We are concerned with a class of ellipsoidal sets (ellipsoids and elliptic cylinders) 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 compact semialgebraic set F. Then we derive a numerically tractable enclosing ellipsoidal set of F 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. Some numerical results on polynomial optimization problems and the sensor network localization problem are presented. This talk is based on a recent joint work with Makoto Yamashita.
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