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

Loading next page...
 
/lp/springer_journal/on-the-conjecture-by-demyanov-ryabova-in-converting-finite-exhausters-KlkjORaUap
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

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off