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Symplectic geometry of constrained optimization

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Abstract

In this paper, we discuss geometric structures related to the Lagrange multipliers rule. The practical goal is to explain how to compute or estimate the Morse index of the second variation. Symplectic geometry allows one to effectively do it even for very degenerate problems with complicated constraints. The main geometric and analytic tool is an appropriately rearranged Maslov index. We try to emphasize the geometric framework and omit analytic routine. Proofs are often replaced with informal explanations, but a well-trained mathematician will easily rewrite them in a conventional way. We believe that Vladimir Arnold would approve of such an attitude.

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Authors and Affiliations

  1. PSI RAS, ul. Petra I 4a, Pereslavl-Zalessky, 152020, Russia

    Andrey A. Agrachev

  2. SISSA, via Bonomea 265, Trieste, 34136, Italy

    Andrey A. Agrachev & Ivan Yu. Beschastnyi

Authors
  1. Andrey A. Agrachev
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  2. Ivan Yu. Beschastnyi
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Correspondence to Andrey A. Agrachev.

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Agrachev, A.A., Beschastnyi, I.Y. Symplectic geometry of constrained optimization. Regul. Chaot. Dyn. 22, 750–770 (2017). https://doi.org/10.1134/S1560354717060119

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  • DOI: https://doi.org/10.1134/S1560354717060119

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