The Demyanov-Ryabova conjecture is false

It was conjectured by Vladimir Demyanov and Julia Ryabova in 2011 that the minimal cycle in the sequence obtained via repeated application of Demyanov converter to a finite family of polytopes is at most two. We construct a counterexample for which the minimal cycle has length 4.Comment: 6 figure

The Demyanov–Ryabova conjecture is false

It was conjectured by Demyanov and Ryabova (Discrete Contin Dyn Syst 31(4):1273–1292, 2011) that the minimal cycle in the sequence obtained via repeated application of the Demyanov converter to a finite family of polytopes is at most two. We construct a counterexample for which the minimal cycle has length 4

Variational Analysis Down Under Open Problem Session

© 2018, Springer Science+Business Media, LLC, part of Springer Nature. We state the problems discussed in the open problem session at Variational Analysis Down Under conference held in honour of Prof. Asen Dontchev on 19–21 February 2018 at Federation University Australia

Geometry in structured optimisation problems

In this thesis, we start by providing some background knowledge on importance of convex analysis. Then, we will be looking at the Demyanov-Ryabova conjecture. This conjecture claims that if we convert between finite families of upper and lower exhausters with the given convertor function, the process will reach a cycle of length at most two. We will show that the conjecture is true in the afflinely independent special case, and also provide an equivalent algebraic reformulation of the conjecture. After that, we will generalise the outer subdifferential construction for max type functions to pointwise minima of regular Lipschitz functions. We will also answer an open question about the relation between the outer subdifferential of the support of a regular function and the end set of its subdifferential. Lastly, we will address the question of what kind of dimensional patterns are possible for the faces of general closed convex sets.  We show that for any finite increasing sequence of positive integers, there exist convex compact sets which only have faces with dimensions from  this prescribed  sequence. We will also discuss another approach to dimensionality by considering  the dimension of the union of all faces of the same dimension. We will demonstrate that the problems arising from this approach are highly nontrivial by providing some examples of convex sets where the sets of extreme points have fractal dimensions