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On the convexity of a class of quadratic mappings and its application to the problem of finding the smallest ball enclosing a given intersection of balls

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Abstract

We consider the outer approximation problem of finding a minimum radius ball enclosing a given intersection of at most n − 1 balls in \({\mathbb{R}^n}\) . We show that if the aforementioned intersection has a nonempty interior, then the problem reduces to minimizing a convex quadratic function over the unit simplex. This result is established by using convexity and representation theorems for a class of quadratic mappings. As a byproduct of our analysis, we show that a class of nonconvex quadratic problems admits a tight semidefinite relaxation.

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Authors and Affiliations

  1. Department of Industrial Engineering and Management, Technion—Israel Institute of Technology, Technion city, Haifa, 32000, Israel

    Amir Beck

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  1. Amir Beck
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Beck, A. On the convexity of a class of quadratic mappings and its application to the problem of finding the smallest ball enclosing a given intersection of balls. J Glob Optim 39, 113–126 (2007). https://doi.org/10.1007/s10898-006-9127-8

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  • DOI: https://doi.org/10.1007/s10898-006-9127-8

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