Abstract
In this work we are interested in the Demyanov–Ryabova conjecture for a finite family of polytopes. The conjecture asserts that after a finite number of iterations (successive dualizations), either a 1-cycle or a 2-cycle eventually comes up. In this work we establish a strong version of this conjecture under the assumption that the initial family contains “enough minimal polytopes” whose extreme points are “well placed”.
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References
Demyanov, V.F., Ryabova, J.A.: Exhausters, coexhausters and converters in nonsmooth analysis. Discrete Contin. Dyn Syst. 31(4), 1273–1292 (2011)
Demyanov, V.F., et al.: Negladkie zadachi teorii optimizatsii i upravleniya. Leningrad University (1982)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of convex analysis, Grundlehren Text Editions. Springer (2001)
Phelps, R.R.: Convex functions, monotone operators and differentiability, Second, Lectures Notes in Mathematics, vol. 1364. Springer, Berlin (1993)
Pshenichnyi, B.N.: Convex Analysis and Extremal Problems. Moscou, Nauka (1980)
Rockafellar, T.R.: Convex Analysis Princeton Mathematical Series, No. 28. N.J, Princeton (1970)
Sang, T.: On the conjecture by Demyanov-Ryabova in converting finite exhausters. J Optim Theory Appl (2017). doi:10.1007/s10957-017-1141-0
Acknowledgments
The authors thank Vera Roshchina (RMIT, Melbourne) for introducing to them the conjecture. They also thank Abderrahim Hantoute (CMM, University of Chile) and Bernard Baillon (University Paris 1) for useful discussions. Major part of this work has been accomplished during a research visit of the second author to the Department of Mathematical Engineering of the University of Chile (October 2016 and April 2017) and of the first author to the Laboratory of Mathematics of the University of Franche-Comté in Besançon (December 2016). The authors thank their hosts for hospitality.
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Aris Daniilidis research was supported by the grants ECOS C14E06, BASAL PFB-03, FONDECYT 1171854 (Chile) and MTM2014-59179-C2-1-P (MINECO-ERDF, Spain and EU). Colin Petitjean research was supported by the grant ECOS-Sud C14E06 (France).
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Daniilidis, A., Petitjean, C. A Partial Answer to the Demyanov-Ryabova Conjecture. Set-Valued Var. Anal 26, 143–157 (2018). https://doi.org/10.1007/s11228-017-0439-2
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DOI: https://doi.org/10.1007/s11228-017-0439-2