Diameter and Curvature: Intriguing Analogies

Deza, Antoine; Terlaky, Tamás; Xie, Feng; Zinchenko, Yuriy

doi:10.1016/j.endm.2008.06.044

Cited by 3 publications

(1 citation statement)

References 19 publications

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“…An analogous behavior is known for both the diameter of a polytope [4] and the total geometric curvature of the central path [3,9,10]. The lower bound Ω(n) for the Sonnevend curvature is also analogous to the worst-case lower bound known for the geometric curvature [3,9,10].…”

Section: Discussionmentioning

confidence: 86%

An Analogue of the Klee–Walkup Result for Sonnevend’s Curvature of the Central Path

Mut

Terlaky

2015

J Optim Theory Appl

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For linear optimization problems, we consider a curvature integral of the central path which was first introduced by Sonnevend et al. (Math Progr 52:527-553, 1991). Our main result states that in order to establish an upper bound for the total Sonnevend curvature of the central path of a polytope, it is sufficient to consider only the case where the number of inequalities is twice as many as the number of variables. This also implies that the worst cases of linear optimization problems for certain pathfollowing interior-point algorithms can be reconstructed for the aforementioned case. As a by-product, our construction yields an asymptotically worst-case lower bound in the order of magnitude of the number of inequalities for Sonnevend's curvature. Our research is motivated by the work of Deza et al. (Electron Notes Discrete Math 31:221-225, 2008) for the geometric curvature of the central path, which is analogous to the Klee-Walkup result for the diameter of a polytope.Mathematics Subject Classification 52B05 · 65K05 · 68Q25 · 90C05 · 90C51 · 90C60 Communicated by Michael Patriksson.

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Section: Discussionmentioning

confidence: 86%

An Analogue of the Klee–Walkup Result for Sonnevend’s Curvature of the Central Path

Mut

Terlaky

2015

J Optim Theory Appl

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Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes

Terlaky

2013

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Rejoinder on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes

Santos

2013

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To foresee the future we must consult the past Hiriart-Urruty reminds us of something I completely agree; in order to advance forward we need to look at the past and understand what others did. In this respect, let me mention that (as is hinted in Sect. 4.2) my counterexamples to the (bounded) Hirsch conjecture (F. Santos 2012) use two ideas from the 45-year old paper by Klee and Walkup (1967). (Unless otherwise stated, all references in this rejoinder refer to the bibliography in the main paper). On the one hand, the Strong d-step Lemma (Theorem 4.8) is an extension of the original d-step lemma of Klee and Walkup (Lemma 4.7). On the other hand, the initial constructions of 5-spindles of length 6 were inspired by the understanding of the Klee-Walkup unbounded non-Hirsch polyhedron in terms of the Cayley trick (see Fig. 11 in Kim and Santos 2009). However, I have to confess I have more than once been guilty of the sin that Hiriart-Urruty describes as everybody writes, nobody reads. And certainly, I agree with Hiriart-Urruty and Poincaré that there are no solved problems. The other three commentators, De Loera, Eisenbrad, and Terlaky, devote most of their commentaries to some of the things that I announce in the Introduction that I do not cover in my paper. I have to thank them for the effort of putting in a few pages so much interesting information. In a sense, I am glad not to have covered these topics, since that made them have to cover themselves, which they have done very nicely. Let me just add a couple of comments myself.

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Diameter and Curvature: Intriguing Analogies

Cited by 3 publications

References 19 publications

An Analogue of the Klee–Walkup Result for Sonnevend’s Curvature of the Central Path

An Analogue of the Klee–Walkup Result for Sonnevend’s Curvature of the Central Path

Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes

Rejoinder on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes

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