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

In this manuscript we provide an exact solution to the maxmin problem <inline-formula> <math display="inline"> <semantics> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo>∥</mo> <mi>A</mi> <mi>x</mi> <mo>∥</mo> </mrow> </semantics> </math> </inline-formula> subject to <inline-formula> <math display="inline"> <semantics> <mrow> <mo>∥</mo> <mi>B</mi> <mi>x</mi> <mo>∥</mo> <mo>&#8804;</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, where <i>A</i> and <i>B</i> are real matrices. This problem comes from a remodeling of <inline-formula> <math display="inline"> <semantics> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo>∥</mo> <mi>A</mi> <mi>x</mi> <mo>∥</mo> </mrow> </semantics> </math> </inline-formula> subject to <inline-formula> <math display="inline"> <semantics> <mrow> <mo movablelimits="true" form="prefix">min</mo> <mo>∥</mo> <mi>B</mi> <mi>x</mi> <mo>∥</mo> </mrow> </semantics> </math> </inline-formula>, 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo>∥</mo> <mi>C</mi> <mi>x</mi> <mo>∥</mo> </mrow> </semantics> </math> </inline-formula> subject to <inline-formula> <math display="inline"> <semantics> <mrow> <mo>∥</mo> <mi>x</mi> <mo>∥</mo> <mo>&#8804;</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, 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.

Abstract licence: CC BY

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