Exact Solutions to the Maxmin Problem max ||Ax|| Subject to ||Bx||<= 1
Abstract
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
Cite
@article{arxiv.2402.06345,
title = {Exact Solutions to the Maxmin Problem max ||Ax|| Subject to ||Bx||<= 1},
author = {Soledad Moreno-Pulido and Francisco Javier García-Pacheco and Clemente Cobos-Sánchez and Alberto Sánchez-Alzola},
journal= {arXiv preprint arXiv:2402.06345},
year = {2024}
}