Exact Solutions to the Maxmin Problem max parallel to Ax parallel to Subject to parallel to Bx parallel to <= 1

Abstract

In this manuscript we provide an exact solution to the maxmin problem max parallel to Ax parallel to subject to parallel to Bx parallel to <= 1, where A and B are real matrices. This problem comes from a remodeling of max parallel to Ax parallel to subject to min parallel to Bx parallel to, 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