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On the Conjecture by Demyanov–Ryabova in Converting Finite Exhausters

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  • Tian Sang

    (RMIT University)

Abstract

The Demyanov–Ryabova conjecture is a geometric problem originating from duality relations between nonconvex objects. Given a finite collection of polytopes, one obtains its dual collection as convex hulls of the maximal facet of sets in the original collection, for each direction in the space (thus constructing upper convex representations of positively homogeneous functions from lower ones and, vice versa, via Minkowski duality). It is conjectured that an iterative application of this conversion procedure to finite families of polytopes results in a cycle of length at most two. We prove a special case of the conjecture assuming an affine independence condition on the vertices of polytopes in the collection. We also obtain a purely combinatorial reformulation of the conjecture.

Suggested Citation

  • Tian Sang, 2017. "On the Conjecture by Demyanov–Ryabova in Converting Finite Exhausters," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 712-727, September.
    • Handle: RePEc:spr:joptap:v:174:y:2017:i:3:d:10.1007_s10957-017-1141-0
    DOI: 10.1007/s10957-017-1141-0
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    References listed on IDEAS

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    1. M. Abbasov & V. Demyanov, 2013. "Proper and adjoint exhausters in nonsmooth analysis: optimality conditions," Journal of Global Optimization, Springer, vol. 56(2), pages 569-585, June.
    2. Jerzy Grzybowski & Diethard Pallaschke & Ryszard Urbański, 2010. "Reduction of finite exhausters," Journal of Global Optimization, Springer, vol. 46(4), pages 589-601, April.
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    Cited by:

    1. Hoa T. Bui & Scott B. Lindstrom & Vera Roshchina, 2019. "Variational Analysis Down Under Open Problem Session," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 430-437, July.

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