Cutting-planes for weakly-coupled 0/1 second order cone programs

Drewes, Sarah; Pokutta, Sebastian

doi:10.1016/j.endm.2010.05.093

Cited by 9 publications

(11 citation statements)

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“…While MILPs offer an incredible representation power, various optimization problems involving risk constraints and discrete decisions give rise to MICPs. Robust optimization and stochastic programming paradigms, or more broadly decision making under uncertainty domain, encompasses many examples of MICPs such as portfolio optimization with fixed transaction costs in finance [34,52], and stochastic joint location-inventory models [4]. Moreover, the most powerful relaxations to many combinatorial optimization problems are based on conic (in particular semidefinite) relaxations (see [35] for a survey on this topic).…”

Section: Introductionmentioning

confidence: 99%

On Minimal Valid Inequalities for Mixed Integer Conic Programs

Kılınç-Karzan

2016

Mathematics of OR

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We study disjunctive conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe the convex hull. We study the properties of K-minimal inequalities by establishing algebraic necessary conditions for an inequality to be K-minimal. This characterization leads to a broader algebraically defined class of Ksublinear inequalities. We establish a close connection between K-sublinear inequalities and the support functions of sets with a particular structure. This connection results in practical ways of showing that a given inequality is K-sublinear and K-minimal.Our framework generalizes some of the results from the mixed integer linear case. It is well known that the minimal inequalities for mixed integer linear programs are generated by sublinear (positively homogeneous, subadditive and convex) functions that are also piecewise linear. This result is easily recovered by our analysis. Whenever possible we highlight the connections to the existing literature. However, our study unveils that such a cut generating function view treating the data associated with each individual variable independently is not possible in the case of general cones other than nonnegative orthant, even when the cone involved is the Lorentz cone.

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Section: Introductionmentioning

confidence: 99%

On Minimal Valid Inequalities for Mixed Integer Conic Programs

Kılınç-Karzan

2016

Mathematics of OR

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“…Finally suppose (−a + cone{c }) ∩ −L n = ∅, and let θ ≥ 0 be such that −a + θc 2 ∈ −L n . Then using (15), In the next result we show that the inequality (5) is sufficient to describe conv(C 1 ∪ C 2 ) when conditions (11) and (12) hold.…”

Section: The Main Resultsmentioning

confidence: 79%

“…Disjunctive cuts were introduced by Balas in the context of mixed-integer linear programming [3] and have since been the cornerstone of theoretical and practical achievements in integer programming. There has been a lot of recent interest in extending disjunctive cutting-plane theory from the domain of mixed-integer linear programming to that of mixed-integer conic programming [2,7,9,11,12,18]. Kılınç-Karzan [13] studied minimal valid linear inequalities for general disjunctive conic sets and showed that these are sufficient to describe the associated closed convex hull under a mild technical assumption.…”

Section: Introductionmentioning

confidence: 99%

Disjunctive cuts for cross-sections of the second-order cone

Yıldız

Cornuéjols

2015

Operations Research Letters

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In this paper we study general two-term disjunctions on affine cross-sections of the secondorder cone. Under some mild assumptions, we derive a closed-form expression for a convex inequality that is valid for such a disjunctive set, and we show that this inequality is sufficient to characterize the closed convex hull of all two-term disjunctions on ellipsoids and paraboloids and a wide class of two-term disjunctions-including split disjunctions-on hyperboloids. Our approach relies on the work of Kılınç-Karzan and Yıldız which considers general two-term disjunctions on the second-order cone.

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“…SOCP is a convex optimization problem, and its solution can be computed with a primal-dual interior-point method [2,38]. MISOCP can be solved globally with, e.g., a branch-and-bound method, because its continuous relaxation is SOCP; see, e.g., [3,13,37] for more details. Applications of MISOCP in engineering can be found in, e.g., [20,21].…”

Section: Introductionmentioning

confidence: 99%