Exact Solutions to the Maxmin Problem max? Ax ? Subject to ? Bx ??1

Soledad Moreno-Pulido, Francisco Javier Garcia-Pacheco, Clemente Cobos-Sanchez and Alberto Sanchez-Alzola
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Soledad Moreno-Pulido: Department of Mathematics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
Francisco Javier Garcia-Pacheco: Department of Mathematics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
Clemente Cobos-Sanchez: Department of Electronics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
Alberto Sanchez-Alzola: Department of Statistics and Operation Research, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain

Mathematics, 2020, vol. 8, issue 1, 1-25

Abstract: 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.

Keywords: maxmin; supporting vector; matrix norm; TMS coil; optimal geolocation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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