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 ||Ax|| subject to ||Bx||<= 1, where A and B are real matrices. This problem comes from a remodeling of max ||Ax|| subject to min ||Bx||, because the latter problem has no solution. Our mathematical method comes from the Abs...

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Published inarXiv.org
Main Authors Moreno-Pulido, Soledad, García-Pacheco, Francisco Javier, Cobos-Sánchez, Clemente, Sánchez-Alzola, Alberto
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.02.2024
Subjects
Exact solutions
Mathematical analysis
Mathematics - Functional Analysis
Mathematics - Mathematical Physics
Mathematics - Optimization and Control
Physics - Mathematical Physics
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Summary:In this manuscript we provide an exact solution to the maxmin problem max ||Ax|| subject to ||Bx||<= 1, where A and B are real matrices. This problem comes from a remodeling of max ||Ax|| subject to min ||Bx||, 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 parallel to Cx parallel to subject to parallel to x parallel to <= 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
Bibliography:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:2331-8422
DOI:10.48550/arxiv.2402.06345