534 research outputs found
Facially exposed cones are not always nice
We address the conjecture proposed by Gabor Pataki that every facially
exposed cone is nice. We show that the conjecture is true in the
three-dimensional case, however, there exists a four-dimensional counterexample
of a cone that is facially exposed but is not nice
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
Directed Subdifferentiable Functions and the Directed Subdifferential without Delta-Convex Structure
We show that the directed subdifferential introduced for differences of
convex (delta-convex, DC) functions by Baier and Farkhi can be constructed from
the directional derivative without using any information on the DC structure of
the function. The new definition extends to a more general class of functions,
which includes Lipschitz functions definable on o-minimal structure and
quasidifferentiable functions.Comment: 30 pages, 3 figure
Applying Metric Regularity to Compute a Condition Measure of a Smoothing Algorithm for Matrix Games
We develop an approach of variational analysis and generalized
differentiation to conditioning issues for two-person zero-sum matrix games.
Our major results establish precise relationships between a certain condition
measure of the smoothing first-order algorithm proposed by Gilpin et al.
[Proceedings of the 23rd AAAI Conference (2008) pp. 75-82] and the exact bound
of metric regularity for an associated set-valued mapping. In this way we
compute the aforementioned condition measure in terms of the initial matrix
game data
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