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

Author

Listed:
  • Soledad Moreno-Pulido

    (Department of Mathematics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
    These authors contributed equally to this work.)

  • Francisco Javier Garcia-Pacheco

    (Department of Mathematics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
    These authors contributed equally to this work.)

  • Clemente Cobos-Sanchez

    (Department of Electronics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
    These authors contributed equally to this work.)

  • Alberto Sanchez-Alzola

    (Department of Statistics and Operation Research, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
    These authors contributed equally to this work.)

Abstract

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.

Suggested Citation

  • Soledad Moreno-Pulido & Francisco Javier Garcia-Pacheco & Clemente Cobos-Sanchez & Alberto Sanchez-Alzola, 2020. "Exact Solutions to the Maxmin Problem max‖ Ax ‖ Subject to ‖ Bx ‖≤1," Mathematics, MDPI, vol. 8(1), pages 1-25, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:85-:d:305251
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    References listed on IDEAS

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    1. Yuta Susowake & Hasan Masrur & Tetsuya Yabiku & Tomonobu Senjyu & Abdul Motin Howlader & Mamdouh Abdel-Akher & Ashraf M. Hemeida, 2019. "A Multi-Objective Optimization Approach towards a Proposed Smart Apartment with Demand-Response in Japan," Energies, MDPI, vol. 13(1), pages 1-14, December.
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