An Algorithm for Finding the Generalized Chebyshev Center of Sets Defined via Their Support Functions
This paper is dedicated to an optimization problem. Let A, B ⊂ R n be compact convex sets. Consider the minimal number t 0 > 0 such that t 0B covers A after a shift to a vector x 0 ∈ R n. The goal is to find t 0 and x 0 . In the special case of B being a unit ball centered at zero, x 0 and t 0 ar...
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| Published in: | Automation and remote control Vol. 85; no. 6; pp. 598 - 609 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
01-06-2024
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| Online Access: | Get full text |
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| Summary: | This paper is dedicated to an optimization problem. Let A, B ⊂ R n be compact convex sets. Consider the minimal number t 0 > 0 such that t 0B covers A after a shift to a vector x 0 ∈ R n. The goal is to find t 0 and x 0 . In the special case of B being a unit ball centered at zero, x 0 and t 0 are known as the Chebyshev center and the Chebyshev radius of A. This paper focuses on the case in which A and B are defined with their black-box support functions. An algorithm for solving such problems efficiently is suggested. The algorithm has a superlinear convergence rate, and it can solve hundred-dimensional test problems in a reasonable time, but some additional conditions on A and B are required to guarantee the presence of convergence. Additionally, the behavior of the algorithm for a simple special case is investigated, which leads to a number of theoretical results. Perturbations of this special case are also studied. |
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| ISSN: | 0005-1179 1608-3032 |
| DOI: | 10.31857/S0005117924060053 |