On the Conjecture by Demyanov–Ryabova in Converting Finite Exhausters

On the Conjecture by Demyanov–Ryabova in Converting Finite Exhausters 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Optimization Theory and Applications Springer Journals

On the Conjecture by Demyanov–Ryabova in Converting Finite Exhausters

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operations Research/Decision Theory
ISSN
0022-3239
eISSN
1573-2878
D.O.I.
10.1007/s10957-017-1141-0
Publisher site
See Article on Publisher Site

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.

Journal

Journal of Optimization Theory and ApplicationsSpringer Journals

Published: Jul 18, 2017

References

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