Generalized mixed integer rounding inequalities: facets for infinite group polyhedra

Kianfar, Kiavash; Fathi, Yahya

doi:10.1007/s10107-008-0216-y

Cited by 22 publications

(25 citation statements)

References 11 publications

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“…a i /α ∈ Z; see [4,5,10,13,16]. Now consider the mixed-integer knapsack set with a single continuous variable and upper bounds on the integer variables…”

Section: Mingling Inequalities: a Brief Reviewmentioning

confidence: 99%

“…The n-step MIR function was introduced in [10] as a tool for producing n-step MIR inequalities for a set that is slightly different from K ≥ , i.e., Y = x ∈ Z N + :…”

Section: N-step Minglingmentioning

confidence: 99%

“…In another direction, for a base mixed-integer constraint Kianfar and Fathi [10] introduce a different generalization of MIR called the n-step MIR inequalities. These inequalities are obtained by applying periodic n-step MIR functions [10] on the coefficients of a base inequality.…”

Section: Introductionmentioning

confidence: 99%

“…These inequalities are obtained by applying periodic n-step MIR functions [10] on the coefficients of a base inequality. Kianfar and Fathi [10,11] show that they are facet-defining for the infinite and finite group problems [6][7][8][9] and also for certain single-constraint mixed integer polyhedra. n-step MIR inequalities generalize the 2-step MIR inequalities of Dash and Günlük [5].…”

Section: Introductionmentioning

confidence: 99%

“…For n ≥ 3, our results present new facets for the mixed integer knapsack set. We also derive conditions under which the n-step MIR inequalities of [10] define facets for the mixed-integer knapsack set as a special case. In addition, we show that n-step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for a mixed-integer knapsack set.…”

Section: Introductionmentioning

confidence: 99%

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n-step mingling inequalities: new facets for the mixed-integer knapsack set

Atamtürk

Kianfar

2010

Math. Program.

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The n-step mixed integer rounding (MIR) inequalities of Kianfar and Fathi (Math Program 120(2):313-346, 2009) are valid inequalities for the mixedinteger knapsack set that are derived by using periodic n-step MIR functions and define facets for group problems. The mingling and 2-step mingling inequalities of Atamtürk and Günlük (Math Program 123(2):315-338, 2010) are also derived based on MIR and they incorporate upper bounds on the integer variables. It has been shown that these inequalities are facet-defining for the mixed integer knapsack set under certain conditions and generalize several well-known valid inequalities. In this paper, we introduce new classes of valid inequalities for the mixed-integer knapsack set with bounded integer variables, which we call n-step mingling inequalities (for positive integer n). These inequalities incorporate upper bounds on integer variables into nstep MIR and, therefore, unify the concepts of n-step MIR and mingling in a single class of inequalities. Furthermore, we show that for each n, the n-step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions. For n = 2, we extend the results of Atamtürk and Günlük on facet-defining properties of 2-step mingling inequalities. For n ≥ 3, we present new facets for the mixed integer knapsack set. As a special case we also derive conditions under which the n-step MIR inequalities define facets for the mixed integer knapsack set. In addition, we show that A. Atamtürk

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“…a i /α ∈ Z; see [4,5,10,13,16]. Now consider the mixed-integer knapsack set with a single continuous variable and upper bounds on the integer variables…”

Section: Mingling Inequalities: a Brief Reviewmentioning

confidence: 99%

“…The n-step MIR function was introduced in [10] as a tool for producing n-step MIR inequalities for a set that is slightly different from K ≥ , i.e., Y = x ∈ Z N + :…”

Section: N-step Minglingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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n-step mingling inequalities: new facets for the mixed-integer knapsack set

Atamtürk

Kianfar

2010

Math. Program.

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Inequalities from Group Relaxations

Richard

2011

Wiley Encyclopedia of Operations Research and Management Science

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This article provides a review of the group‐theoretic approach to the generation of cutting planes in mixed integer programming. After motivating the notion of corner relaxation graphically, we give a definition of master group problems. We then present a hierarchy of valid inequalities for master group problems that can be used as cutting planes for mixed integer programs. We describe next various procedures that can be used to obtain the “strongest” valid inequalities for master group problems. We conclude by commenting on the computational possibilities of group‐theoretic cuts in mixed integer programming.

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Valid inequalities and facets for multi‐module survivable network design problem

Luo

Kianfar

2022

Networks

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Capacitated survivable network design (SND), that is, designing a network that can survive edge failures with minimum flow-routing plus capacity-installation cost, is a fundamental problem in network science and its applications. In this article, we study a highly applicable form of the SND, referred to as the multi-module SND (MM-SND), in which transmission capacities on edges can be sum of integer multiples of differently sized capacity modules. We formulate MM-SND as mixed integer programs, where existing structures of the network (p-cycles) are used to protect edge failures. We derive valid inequalities for MM-SND through polyhedral analysis and show that the valid inequalities previously developed in the literature for the single-module SND are special cases of these inequalities. Furthermore, we show that these inequalities define facets for the convex hull of MM-SND in many cases. Our computational results show that these inequalities are very effective in solving MM-SND problem instances.

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Generalized mixed integer rounding inequalities: facets for infinite group polyhedra

Cited by 22 publications

References 11 publications

n-step mingling inequalities: new facets for the mixed-integer knapsack set

n-step mingling inequalities: new facets for the mixed-integer knapsack set

Inequalities from Group Relaxations

Valid inequalities and facets for multi‐module survivable network design problem

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