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21
A strong conic quadratic reformulation for machinejob assignment with controllable processing times
 OPERATIONS RESEARCH LETTERS
, 2009
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Lifting for conic mixedinteger programming
, 2011
"... Lifting is a procedure for deriving valid inequalities for mixedinteger sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to sol ..."
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Cited by 15 (4 self)
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Lifting is a procedure for deriving valid inequalities for mixedinteger sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branchandcut algorithms. Here we generalize the theory of lifting to conic integer programming, i.e., integer programs with conic constraints. We show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lowerdimensional restrictions. In order to simplify the computations, we also discuss sequenceindependent lifting for conic integer programs. When the cones are restricted to nonnegative orthants, conic lifting reduces to the lifting for linear integer programming as one may expect.
A Conic Integer Programming Approach to Stochastic Joint LocationInventory Problems
, 2012
"... We study several joint facility location and inventory management problems with stochastic retailer demand. In particular, we consider cases with uncapacitated facilities, capacitated facilities, correlated retailer demand, stochastic lead times, and multicommodities. We show how to formulate these ..."
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Cited by 6 (0 self)
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We study several joint facility location and inventory management problems with stochastic retailer demand. In particular, we consider cases with uncapacitated facilities, capacitated facilities, correlated retailer demand, stochastic lead times, and multicommodities. We show how to formulate these problems as conic quadratic mixedinteger problems. Valid inequalities, including extended polymatroid and extended cover cuts, are added to strengthen the formulations and improve the computational results. Compared to the existing modeling and solution methods, the new conic integer programming approach not only provides a more general modeling framework but also leads to fast solution times in general.
On Families of Quadratic Surfaces Having Fixed Intersections with Two Hyperplanes
, 2011
"... We investigate families of quadrics that have fixed intersections with two given hyperplanes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter. In particular we show how the quadrics are ..."
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Cited by 4 (0 self)
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We investigate families of quadrics that have fixed intersections with two given hyperplanes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter. In particular we show how the quadrics are transformed as the parameter changes. This research was motivated by an application in mixedinteger conic optimization. In that application we aimed to characterize the convex hull of the union of the intersections of an ellipsoid with two halfspaces when these intersections are disjunctive sets.
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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Cited by 4 (0 self)
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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.
Using interiorpoint methods within an outer approximation framework for mixedinteger nonlinear programming
 IMAMINLP Issue
"... Abstract. Interiorpoint methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixedinteger nonlinear programming problems via o ..."
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Abstract. Interiorpoint methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixedinteger nonlinear programming problems via outer approximation. However, traditionally, infeasible primaldual interiorpoint methods have had two main perceived deficiencies: (1) lack of infeasibility detection capabilities, and (2) poor performance after a warmstart. In this paper, we propose the exact primaldual penalty approach as a means to overcome these deficiencies. The generality of this approach to handle any change to the problem makes it suitable for the outer approximation framework, where each nonlinear subproblem can differ from the others in the sequence in a variety of ways. Additionally, we examine cases where the nonlinear subproblems take on special forms, namely those of secondorder cone programming problems and semidefinite programming problems. Encouraging numerical results are provided.
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.
A geometric approach to cutgenerating functions
, 2015
"... The cuttingplane approach to integer programming was initiated more that 40 years ago: Gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and Balas introduced intersection cuts for the corner polyhedron. This line of research was left dormant for several ..."
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The cuttingplane approach to integer programming was initiated more that 40 years ago: Gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and Balas introduced intersection cuts for the corner polyhedron. This line of research was left dormant for several decades until relatively recently, when a paper of Andersen, Louveaux, Weismantel and Wolsey generated renewed interest in the corner polyhedron and intersection cuts. Recent developments rely on tools drawn from convex analysis, geometry and number theory, and constitute an elegant bridge between these areas and integer programming. We survey these results and highlight recent breakthroughs in this area.