Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1
In this manuscript we provide an exact solution to the maxmin problem max ∥ A x ∥ subject to ∥ B x ∥ ≤ 1 , where A and B are real matrices. This problem comes from a remodeling of max ∥ A x ∥ subject to min ∥ B x ∥ , because the latter problem has no solution. Our mathematical method comes from the...
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Published in | Mathematics (Basel) Vol. 8; no. 1; p. 85 |
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Main Authors | , , , |
Format | Journal Article |
Language | English |
Published |
Basel
MDPI AG
01.01.2020
MDPI |
Subjects | |
Online Access | Get full text |
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Summary: | In this manuscript we provide an exact solution to the maxmin problem max ∥ A x ∥ subject to ∥ B x ∥ ≤ 1 , where A and B are real matrices. This problem comes from a remodeling of max ∥ A x ∥ subject to min ∥ B x ∥ , because the latter problem has no solution. Our mathematical method comes from the Abstract Operator Theory, whose strong machinery allows us to reduce the first problem to max ∥ C x ∥ subject to ∥ x ∥ ≤ 1 , which can be solved exactly by relying on supporting vectors. Finally, as appendices, we provide two applications of our solution: first, we construct a truly optimal minimum stored-energy Transcranian Magnetic Stimulation (TMS) coil, and second, we find an optimal geolocation involving statistical variables. |
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ISSN: | 2227-7390 2227-7390 |
DOI: | 10.3390/math8010085 |