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...

Full description

Saved in:
Bibliographic Details
Published inMathematics (Basel) Vol. 8; no. 1; p. 85
Main Authors Moreno-Pulido, Soledad, Garcia-Pacheco, Francisco Javier, Cobos-Sanchez, Clemente, Sanchez-Alzola, Alberto
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.01.2020
MDPI
Subjects
Banach spaces
Coils
Exact solutions
Food science
Mathematical analysis
Mathematical functions
Mathematics
matrix norm
maxmin
optimal geolocation
Optimization
Principal components analysis
supporting vector
tms coil
Vectors (mathematics)
maxmin
TMS coil
49J30
47L90
optimal geolocation MSC: 47L05
maxmin supporting vector matrix norm TMS coil optimal geolocation MSC: 47L05
90B50
supporting vector
matrix norm
Online AccessGet full text

Cover

More Information
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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math8010085