Intersection Cuts for Mixed Integer Conic Quadratic Sets

Andersen, Kim Allan; Jensen, Anders Nedergaard

doi:10.1007/978-3-642-36694-9_4

Cited by 29 publications

(56 citation statements)

References 11 publications

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“…In contrast, if K is a general closed convex set, K π,π 0 is only closed and convex [14]. However, for special classes of K, we can characterize the nonlinear split cuts that need to be added to K to obtain K π,π 0 [1,4,5,14,19,21]. For instance, the following proposition from [21] characterizes split cuts for conic quadratic sets of the form…”

Section: Notation and Previous Workmentioning

confidence: 99%

“…Proposition 1. Let B ∈ R n×n be an invertible matrix, c ∈ R n , (π, π 0 ) ∈ Z n × Z, and C be as defined in (1). If π T c (π 0 , π 0 + 1), then C π,π 0 = C. Otherwise, there exist B ∈ R n×n andc ∈ R n such that…”

Section: Notation and Previous Workmentioning

confidence: 99%

“…The study of split cuts for MINLP is still much more limited than for MILP; however, there has been significant work on the computational use of split cuts in MINLP [9,11,16,18,25] and a recent surge of theoretical developments [1,2,4,5,14,19,21,22]. In particular, several formulas for split cuts for Mixed Integer Conic Quadratic Programming (MICQP) have been recently developed [1,4,5,14,21]. While the resulting cuts are strong nonlinear inequalities, adding these cuts to the continuous relaxation of the MICQP can significantly increase its solution time, which could negate the effectiveness of the cuts.…”

Section: Introductionmentioning

confidence: 99%

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Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

Modaresi

Kılınç

Vielma

2015

Operations Research Letters

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We study split cuts and extended formulations for Mixed Integer Conic Quadratic Programming (MICQP) and their relation to Conic Mixed Integer Rounding (CMIR) cuts. We show that CMIR is a linear split cut for the polyhedral portion of an extended formulation of a quadratic set and it can be weaker than the nonlinear split cut of the same quadratic set. However, we also show that families of CMIRs can be significantly stronger than the associated family of nonlinear split cuts.

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Section: Notation and Previous Workmentioning

confidence: 99%

Section: Notation and Previous Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

Modaresi

Kılınç

Vielma

2015

Operations Research Letters

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“…For example, such sets cover the simpler setups commonly studied such as the two-term disjunctions or split disjunctions on regular cones or their cross-sections. The particular case of two-term or split disjunctions on K = L n has recently attracted a lot of attention [1,2,6,7,9,12,15,[23][24][25][26]30].…”

Section: Introductionmentioning

confidence: 99%

On sublinear inequalities for mixed integer conic programs

Kılınç-Karzan

Steffy

2016

Math. Program.

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This paper studies K-sublinear inequalities, a class of inequalities with strong relations to K-minimal inequalities for disjunctive conic sets. We establish a stronger result on the sufficiency of K-sublinear inequalities. That is, we show that when K is the nonnegative orthant or the second-order cone, K-sublinear inequalities together with the original conic constraint are always sufficient for the closed convex hull description of the associated disjunctive conic set. When K is the nonnegative orthant, K-sublinear inequalities are tightly connected to functions that generate cutsso called cut-generating functions. In particular, we introduce the concept of relaxed cut-generating functions and show that each R n + -sublinear inequality is generated by one of these. We then relate the relaxed cut-generating functions to the usual ones studied in the literature. Recently, under a structural assumption, Cornuéjols, Wolsey and Yıldız established the sufficiency of cut-generating functions in terms of generating all nontrivial valid inequalities of disjunctive sets where the underlying cone is nonnegative orthant. We provide an alternate and straightforward proof of this result under the same assumption as a consequence of the sufficiency of R n + -sublinear inequalities and their connection with relaxed cut-generating functions.

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“…Belotti et al [12] give conic cuts for conic quadratic integer optimization. Anderson and Jensen [3] give intersection cuts for conic quadratic mixed-integer sets. Kılınç [30] describes minimal inequalities for conic mixed-integer programs.…”

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confidence: 99%