Piecewise smooth extreme functions are piecewise linear

Summa, Marco Di

doi:10.1007/s10107-018-1330-0

Cited by 3 publications

(3 citation statements)

References 14 publications

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“…A fruitful approach to extremality testing, introduced by Basu et al in Part I of the present series of papers [3], has been to consider the difference function (perturbation) π = π + − π, which allows us to write π + = π + π and π − = π − π. (Recently, Di Summa [9] obtained a breakthrough result on the question of piecewise linearity of extreme functions using this approach.) It is convenient to build a space from the notion of perturbation functions.…”

Section: The Space ππ Of Effective Perturbations π Of a Minimal Valid...mentioning

confidence: 99%

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations

Hildebrand¹,

Köppe²,

Zhou³

2022

Open Journal of Mathematical Optimization

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In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite-and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for the Gomory-Johnson model, in particular for testing extremality of piecewise linear functions whose breakpoints are rational numbers with huge denominators.

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Section: The Space ππ Of Effective Perturbations π Of a Minimal Valid...mentioning

confidence: 99%

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations

Hildebrand¹,

Köppe²,

Zhou³

2022

Open Journal of Mathematical Optimization

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“…For an overview we refer to the surveys [21, sections 19.4-19.6] and [3,4]. For recent developments we refer to [1,9,15]. 1.1.…”

Section: Introductionmentioning

confidence: 99%

All Cyclic Group Facets Inject

Köppe,

Zhou

2018

Preprint

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We give a variant of Basu-Hildebrand-Molinaro's approximation theorem for continuous minimal valid functions for Gomory-Johnson's infinite group problem by piecewise linear two-slope extreme functions [Minimal cutgenerating functions are nearly extreme, IPCO 2016]. Our theorem is for piecewise linear minimal valid functions that have only rational breakpoints (in 1/q Z for some q ∈ N) and that take rational values at the breakpoints. In contrast to Basu et al.'s construction, our construction preserves all function values on 1/q Z. As a corollary, we obtain that every extreme function for the finite group problem on 1/q Z is the restriction of a continuous piecewise linear two-slope extreme function for the infinite group problem with breakpoints on a refinement 1/(M q) Z for some M ∈ N. In combination with Gomory's master theorem [Some Polyhedra related to Combinatorial Problems, Lin. Alg. Appl. 2 (1969), 451-558], this shows that the infinite group problem is the correct master problem for facets (extreme functions) of 1-row group relaxations.

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“…A fruitful approach to extremality testing, introduced by Basu et al in Part I of the present series of papers [3], has been to consider the difference function (perturbation) π = π + − π, which allows us to write π 1 = π + π and π 2 = π − π. (Recently, Di Summa [8] obtained a breakthrough result on the question of piecewise linearity of extreme functions using this approach.) It is convenient to build a space from the notion of perturbation functions.…”

mentioning

confidence: 99%

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

Hildebrand,

Köppe,

Zhou

2018

Preprint

View full text Add to dashboard Cite

show abstract

Piecewise smooth extreme functions are piecewise linear

Cited by 3 publications

References 14 publications

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations

All Cyclic Group Facets Inject

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

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