Next Article in Journal
Comultiplications on the Localized Spheres and Moore Spaces
Previous Article in Journal
Sustainable Closed-Loop Supply Chain Design Problem: A Hybrid Genetic Algorithm Approach
Previous Article in Special Issue
Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations
Open AccessArticle

Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1

1
Department of Mathematics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
2
Department of Electronics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
3
Department of Statistics and Operation Research, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(1), 85; https://doi.org/10.3390/math8010085
Received: 15 October 2019 / Revised: 28 December 2019 / Accepted: 30 December 2019 / Published: 4 January 2020
(This article belongs to the Special Issue Numerical Methods)
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. View Full-Text
Keywords: maxmin; supporting vector; matrix norm; TMS coil; optimal geolocation maxmin; supporting vector; matrix norm; TMS coil; optimal geolocation
Show Figures

Graphical abstract

MDPI and ACS Style

Moreno-Pulido, S.; Garcia-Pacheco, F.J.; Cobos-Sanchez, C.; Sanchez-Alzola, A. Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1. Mathematics 2020, 8, 85.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop