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Two term disjunctions on the secondorder cone
, 2014
"... Balas introduced disjunctive cuts in the 1970s for mixedinteger linear programs. Several recent papers have attempted to extend this work to mixedinteger conic programs. In this paper we study the structure of the convex hull of a twoterm disjunction applied to the secondorder cone, and develop ..."
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Balas introduced disjunctive cuts in the 1970s for mixedinteger linear programs. Several recent papers have attempted to extend this work to mixedinteger conic programs. In this paper we study the structure of the convex hull of a twoterm disjunction applied to the secondorder cone, and develop a methodology to derive closedform 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 twoterm disjunctions on the secondorder cone generalize related results on split cuts by Modaresi, Kılınç, and Vielma, and by Andersen and Jensen.
On Minimal Valid Inequalities for Mixed Integer Conic Programs
"... 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 Kminimal inequalities and show that, under mild assumptions, ..."
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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 Kminimal inequalities and show that, under mild assumptions, these inequalities together with the trivial coneimplied inequalities are sufficient to describe the convex hull. We focus on the properties ofKminimal inequalities by establishing algebraic necessary conditions for an inequality to be Kminimal. This characterization leads to a broader algebraically defined class of Ksublinear inequalities. We demonstrate a close connection between Ksublinear inequalities and the support functions of convex sets with a particular structure. This connection results in practical ways of verifying Ksublinearity and/or Kminimality of inequalities. Our study 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 which are also piecewise linear. Our analysis easily recovers this result. However, in the case of general regular cones other than the nonnegative orthant, our study reveals that such a cutgenerating function view that treats the data associated with each individual variable independently is far from sufficient.
How to Convexify the Intersection of a Second Order Cone and a Nonconvex Quadratic
 MATHEMATICAL PROGRAMMING
, 2014
"... A recent series of papers has examined the extension of disjunctiveprogramming techniques to mixedinteger secondordercone programming. For example, it has been shown—by several authors using different techniques—that the convex hull of the intersection of an ellipsoid, E, and a split disjuncti ..."
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A recent series of papers has examined the extension of disjunctiveprogramming techniques to mixedinteger secondordercone programming. For example, it has been shown—by several authors using different techniques—that the convex hull of the intersection of an ellipsoid, E, and a split disjunction, (l − xj)(xj − u) ≤ 0 with l < u, equals the intersection of E with an additional secondordercone representable (SOCr) set. In this paper, we study more general intersections of the form K ∩ Q and K ∩ Q ∩H, where K is a SOCr cone, Q is a nonconvex cone defined by a single homogeneous quadratic, and H is an affine hyperplane. Under several easytoverify conditions, we derive simple, computable convex relaxations K∩S and K∩S∩H, where S is a SOCr cone. Under further conditions, we prove that these two sets capture precisely the corresponding conic/convex hulls. Our approach unifies and extends previous results, and we illustrate its applicability and generality with many examples.
Disjunctive cuts for crosssections of the secondorder cone
 Operations Research Letters
, 2015
"... Abstract In this paper we provide a unified treatment of general twoterm disjunctions on crosssections of the secondorder cone. We derive a closedform expression for a convex inequality that is valid for such a disjunctive set and show that this inequality is sufficient to characterize the close ..."
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Abstract In this paper we provide a unified treatment of general twoterm disjunctions on crosssections of the secondorder cone. We derive a closedform expression for a convex inequality that is valid for such a disjunctive set and show that this inequality is sufficient to characterize the closed convex hull of all twoterm disjunctions on ellipsoids and paraboloids, and split disjunctions on all crosssections of the secondorder cone. Our approach extends the work of KılınçKarzan and Yıldız on general twoterm disjunctions for the secondorder cone.
Noname manuscript No. (will be inserted by the editor) On Sublinear Inequalities for Mixed Integer Conic Programs
, 2014
"... Abstract This paper studies Ksublinear inequalities, a class of inequalities with strong relations to Kminimal inequalities for disjunctive conic sets. We establish a stronger result on the sufficiency of Ksublinear inequalities. That is, we show that when K is the nonnegative orthant or the seco ..."
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Abstract This paper studies Ksublinear inequalities, a class of inequalities with strong relations to Kminimal inequalities for disjunctive conic sets. We establish a stronger result on the sufficiency of Ksublinear inequalities. That is, we show that when K is the nonnegative orthant or the secondorder cone, Ksublinear 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, Ksublinear inequalities are tightly connected to functions that generate cuts—so called cutgenerating functions. As a consequence of the sufficiency of Rn+sublinear inequalities, we also provide an alternate and straightforward proof of the sufficiency of cutgenerating functions for mixed integer linear programs, a result recently established by Cornuéjols, Wolsey and Yıldız. 1
Cutting Planes for Convex Objective Nonconvex Optimization
, 2013
"... This thesis studies methods for tightening relaxations of optimization problems with convex objective values over a nonconvex domain. A class of linear inequalities obtained by lifting easily obtained valid inequalities is introduced, and it is shown that this class of inequalities is sufficient to ..."
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This thesis studies methods for tightening relaxations of optimization problems with convex objective values over a nonconvex domain. A class of linear inequalities obtained by lifting easily obtained valid inequalities is introduced, and it is shown that this class of inequalities is sufficient to describe the epigraph of a convex and differentiable function over a general domain. In the special case where the objective is a positive definite quadratic function, polynomial time separation procedures using the new class of lifted inequalities are developed for the cases when the domain is the complement of the interior of a polyhedron, a union of polyhedra, or the complement of the interior of an ellipsoid. Extensions for positive semidefinite and indefinite quadratic objectives are also studied. Applications and computational considerations are discussed, and the results from a series of numerical experiments are presented.