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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Authors: | Amir Beck |
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Affiliation: | (1) Department of Industrial Engineering and Management, Technion—Israel Institute of Technology, Technion city, Haifa, 32000, Israel |
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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 . 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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Keywords: | Outer approximation problems Convexity of quadratic mappings Nonconvex quadratic optimization Semidefinite relaxation Strong duality |
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