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The Maximax Minimax Quotient Theorem

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Abstract

We present an optimization problem emerging from optimal control theory and situated at the intersection of fractional programming and linear max-min programming on polytopes. A naïve solution would require solving four nested, possibly nonlinear, optimization problems. Instead, relying on numerous geometric arguments we determine an analytical solution to this problem. In the course of proving our main theorem, we also establish another optimization result stating that the minimum of a specific minimax optimization is located at a vertex of the constraint set.

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Acknowledgements

This work was supported by an Early Stage Innovations grant from NASA’s Space Technology Research Grants Program, grant no. 80NSSC19K0209. This material is partially based upon work supported by the United States Air Force AFRL/SBRK under contract no. FA864921P0123.

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Authors and Affiliations

  1. Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

    Jean-Baptiste Bouvier

  2. Department of Aerospace Engineering and Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

    Melkior Ornik

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  1. Jean-Baptiste Bouvier
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  2. Melkior Ornik
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Correspondence to Jean-Baptiste Bouvier.

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Communicated by Alexander Mitsos.

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Bouvier, JB., Ornik, M. The Maximax Minimax Quotient Theorem. J Optim Theory Appl 192, 1084–1101 (2022). https://doi.org/10.1007/s10957-022-02008-z

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