Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

Modaresi, Sina; Kılınç, Mustafa R.; Vielma, Juan Pablo

doi:10.1016/j.orl.2014.10.006

Cited by 33 publications

(21 citation statements)

References 23 publications

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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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Section: Introductionmentioning

confidence: 99%

On sublinear inequalities for mixed integer conic programs

Kılınç-Karzan

Steffy

2016

Math. Program.

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“…Dadush et al [20] and Andersen and Jensen [1] derived split cuts for ellipsoids and the secondorder cone, respectively. Modaresi et al [26] extended this work on split disjunctions to essentially all cross-sections of the second-order cone, and studied their theoretical and computational relations with extended formulations and conic MIR inequalities in [27]. Belotti et al [10] studied the families of quadratic surfaces having fixed intersections with two given hyperplanes and showed that these families can be described by a single parameter.…”

Section: Introductionmentioning

confidence: 99%

Two-term disjunctions on the second-order cone

Kılınç-Karzan

Yıldız

2015

Math. Program.

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Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we study the structure of the convex hull of a two-term disjunction applied to the second-order cone, and develop a methodology to derive closed-form expressions for convex inequalities describing the resulting convex hull. Our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex, possibly nonlinear, valid inequalities that correspond to these linear inequalities. We identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull. In particular, our results on two-term disjunctions on the second-order cone generalize related results on split cuts by Modaresi, Kılınç, and Vielma, and by Andersen and Jensen.

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“…Therefore, some of the cuts generated using (6) can be viewed as split disjunctive cuts. Significant research has gone into describing split disjunctive cuts (newer implied conic constraints) for conic sections [21,22,23,24,25,26,27]. However, to the best of our knowledge, there is no family of subadditive functions in F L m which have been described in closed form previously.…”

Section: New Family Of Cut-generating Functionsmentioning

confidence: 99%

“…Clearly, x * is feasible for (25). Note now that the dual solution constructed above for (23), when restricted to the y i 0 component is a feasible solution to (26) with objective value α. Thus, we have α ≤ β ≤ π ⊤ x * ≤ α + ε, where the first inequality follows from weak duality to the primal-dual pair (25)(26) and the second inequality follows from fisibility of x * to (25).…”

Section: Lemmamentioning

confidence: 99%

Some cut-generating functions for second-order conic sets

Santana

Dey

2017

Discrete Optimization

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In this paper, we study cut generating functions for conic sets. Our first main result shows that if the conic set is bounded, then cut generating functions for integer linear programs can easily be adapted to give the integer hull of the conic integer program. Then we introduce a new class of cut generating functions which are non-decreasing with respect to second-order cone. We show that, under some minor technical conditions, these functions together with integer linear programming-based functions are sufficient to yield the integer hull of intersections of conic sections in R 2 .

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Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

Cited by 33 publications

References 23 publications

On sublinear inequalities for mixed integer conic programs

On sublinear inequalities for mixed integer conic programs

Two-term disjunctions on the second-order cone

Some cut-generating functions for second-order conic sets

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