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Enclosing ellipsoids and elliptic cylinders of semialgebraic sets and their application to error bounds in polynomial optimization

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Abstract

This paper is concerned with a class of ellipsoidal sets (ellipsoids and elliptic cylinders) in \({\mathbb{R}^m}\) which are determined by a freely chosen m × m 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 \({\mathbb{R}^m}\) . Then we derive a numerically tractable enclosing ellipsoidal set of a given semialgebraic subset of \({\mathbb{R}^m}\) 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 standard SDP relaxation for quadratic optimization problems and a sparse variant of Lasserre’s hierarchy SDP relaxation for polynomial optimization problems. Some numerical results on the sensor network localization problem and polynomial optimization problems are also presented.

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Authors and Affiliations

  1. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo, 152-8552, Japan

    Masakazu Kojima & Makoto Yamashita

Authors
  1. Masakazu Kojima
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  2. Makoto Yamashita
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Corresponding author

Correspondence to Masakazu Kojima.

Additional information

M. Kojima research was partially supported by Grant-in-Aid for Scientific Research (B) 19310096.

M. Yamashita’s research was supported by Grant-in-Aid for Young Scientists (B) 21710148.

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Kojima, M., Yamashita, M. Enclosing ellipsoids and elliptic cylinders of semialgebraic sets and their application to error bounds in polynomial optimization. Math. Program. 138, 333–364 (2013). https://doi.org/10.1007/s10107-012-0515-1

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  • DOI: https://doi.org/10.1007/s10107-012-0515-1

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