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Review of nonlinear mixedinteger and disjunctive programming techniques
 Optimization and Engineering
, 2002
"... This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, OuterApproximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are ex ..."
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Cited by 85 (19 self)
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This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, OuterApproximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.
Mixed Logical/Linear Programming
 Discrete Applied Mathematics
, 1996
"... Mixed logical/linear programming (MLLP) is an extension of mixed integer/linear programming (MILP). It represents the discrete elements of a problem with logical propositions and provides a more natural modeling framework than MILP. It can also have computational advantages, partly because it elimin ..."
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Cited by 38 (10 self)
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Mixed logical/linear programming (MLLP) is an extension of mixed integer/linear programming (MILP). It represents the discrete elements of a problem with logical propositions and provides a more natural modeling framework than MILP. It can also have computational advantages, partly because it eliminates integer variables when they serve no purpose, provides alternatives to the traditional continuous relaxation, and applies logic processing algorithms. This paper surveys previous work and attempts to organize ideas associated with MLLP, some old and some new, into a coherent framework. It articulates potential advantages and disadvantages of MLLP and illustrates some of them with computational experiments. 1 Introduction Mixed logical/linear programming (MLLP) is a general approach to formulating and solving optimization problems that have both discrete and continuous elements. Mixed integer/linear programming (MILP), the traditional approach, is effective in many instances. But it unn...
LiftandProject for Mixed 01 Programming: Recent Progress
, 1999
"... Contents 1 Introduction 2 Disjunctive programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Two basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Compact Representation of the Convex Hull 3 Projection and polarity . . . . . . . . . ..."
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Cited by 18 (1 self)
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Contents 1 Introduction 2 Disjunctive programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Two basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Compact Representation of the Convex Hull 3 Projection and polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Adjacency on the higher dimensional polyhedron . . . . . . . . . . . . . . . . . . 5 3 Sequential Convexication 7 Disjunctive rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Fractionality of intermediate points . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Another Derivation of the Basic Results 12 5 Generating Cuts 13 Deepest cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Cut lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Cut strengthening . . . . . . . .
Logic, Optimization, and Constraint Programming
 INFORMS Journal on Computing
, 2000
"... Because of their complementary strengths, optimization and constraint programming can be profitably merged. Their integration has been the subject of increasing commercial and research activity. This paper summarizes and contrasts the characteristics of the two fields; in particular, how they use ..."
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Cited by 13 (2 self)
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Because of their complementary strengths, optimization and constraint programming can be profitably merged. Their integration has been the subject of increasing commercial and research activity. This paper summarizes and contrasts the characteristics of the two fields; in particular, how they use logical inference in di#erent ways, and how these ways can be combined. It sketches the intellectual background for recent e#orts at integration. In particular, it traces the history of logicbased methods in optimization and the development of constraint programming in artificial intelligence. It concludes with a review of recent research, with emphasis on schemes for integration, relaxation methods, and practical applications. Optimization and constraint programming are beginning to converge, despite their very di#erent origins. Optimization is primarily associated with mathematics and engineering, while constraint programming developed much more recently in the computer science an...
Modeling of Discrete/Continuous Optimization Problems: Characterization and Formulation of Disjunctions and their Relaxations
, 2002
"... Abstract. This paper addresses the relaxations in alternative models for disjunctions, bigM and convex hull model, in order to develop guidelines and insights when formulating MixedInteger NonLinear Programming (MINLP), Generalized Disjunctive Programming (GDP), or hybrid models. Characterization ..."
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Cited by 12 (4 self)
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Abstract. This paper addresses the relaxations in alternative models for disjunctions, bigM and convex hull model, in order to develop guidelines and insights when formulating MixedInteger NonLinear Programming (MINLP), Generalized Disjunctive Programming (GDP), or hybrid models. Characterization and properties are presented for various types of disjunctions. An interesting result is presented for improper disjunctions where results in the continuous space differ from the ones in the mixedinteger space. A cutting plane method is also proposed that avoids the explicit generation of equations and variables of the convex hull. Several examples are presented throughout the paper, as well as a small process synthesis problem, which is solved with the proposed cutting plane method.
BranchandCut for Combinatorial Optimization Problems without Auxiliary Binary Variables
 KNOWLEDGE ENGINEERING REVIEW
, 2001
"... Many optimization problems involve combinatorial constraints on continuous variables. An example of a combinatorial constraint is that at most one variable in a group of nonnegative variables may be positive. Traditionally, in the mathematical programming community, such problems have been modeled a ..."
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Cited by 10 (3 self)
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Many optimization problems involve combinatorial constraints on continuous variables. An example of a combinatorial constraint is that at most one variable in a group of nonnegative variables may be positive. Traditionally, in the mathematical programming community, such problems have been modeled as mixedinteger programs by introducing auxiliary binary variables and additional constraints. Because the number of variables and constraints becomes larger and the combinatorial structure is not used to advantage, these mixedinteger programming models may not be solved satisfactorily, except for small instances. Traditionally, constraint programming approaches to such problems keep and use the combinatorial structure, but do not use linear programming bounds in the search for an optimal solution. Here we present a branchandcut approach that considers the combinatorial constraints without the introduction of binary variables. We review the development of this approach and show how strong constraints can be derived using ideas from polyhedral combinatorics. To illustrate the ideas, we present a production scheduling model that arises in the manufacture of fiber optic cables.
Projection, lifting and extended formulation in integer and combinatorial optimization
 Annals of Operations Research
, 2005
"... Abstract. This is an overview of the significance and main uses of projection, lifting and extended formulation in integer and combinatorial optimization. Its first two sections deal with those basic properties of projection that make it such an effective and useful bridge between problem formulati ..."
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Cited by 9 (0 self)
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Abstract. This is an overview of the significance and main uses of projection, lifting and extended formulation in integer and combinatorial optimization. Its first two sections deal with those basic properties of projection that make it such an effective and useful bridge between problem formulations in different spaces, i.e. different sets of variables. They discuss topics like projection and restriction, the integralitypreserving property of projection, the dimension of projected polyhedra, conditions for facets of a polyhedron to project into facets of its projections, and so on. The next two sections describe the use of projection for comparing the strength of different formulations of the same problem, and for proving the integrality of polyhedra by using extended formulations or lifting. Section 5 deals with disjunctive programming, or optimization over unions of polyhedra, whose most important incarnation are mixed 01 programs and their partial relaxations. It discusses the compact representation of the convex hull of a union of polyhedra through extended formulation, the connection between the projection of the latter and the polar of the convex hull, as well as the sequential convexification of facial disjunctive programs, among them mixed 01 programs, with the related concept of disjunctive rank. Section 6 reviews liftandproject cuts, the construction of cut generating linear programs, and techniques for lifting and for strengthening disjunctive cuts. Section 7 discusses the recently discovered possibility of solving the higher dimensional cut generating linear program without explicitly constructing it, by a sequence of properly chosen pivots in the simplex tableau of the linear programming relaxation. Finally, section 8 deals with different ways of combining cuts with branch and bound, and briefly discusses computational experience with liftandproject cuts.
Venn Sampling: A Novel Prediction Technique for Moving Objects
 In ICDE
, 2005
"... Given a region qR and a future timestamp qT, a “range aggregate ” query estimates the number of objects expected to appear in qR at time qT. Currently the only methods for processing such queries are based on spatiotemporal histograms, which have several serious problems. First, they consume conside ..."
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Cited by 8 (0 self)
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Given a region qR and a future timestamp qT, a “range aggregate ” query estimates the number of objects expected to appear in qR at time qT. Currently the only methods for processing such queries are based on spatiotemporal histograms, which have several serious problems. First, they consume considerable space in order to provide accurate estimation. Second, they incur high evaluation cost. Third, their efficiency continuously deteriorates with time. Fourth, their maintenance requires significant update overhead. Motivated by this, we develop Venn sampling (VS), a novel estimation method optimized for a set of “pivot queries ” that reflect the distribution of actual ones. In particular, given m pivot queries, VS achieves perfect estimation with only O(m) samples, as opposed to O(2 m) required by the current state of the art in workloadaware sampling. Compared with histograms, our technique is much more accurate (given the same space), produces estimates with negligible cost, and does not deteriorate with time. Furthermore, it permits the development of a novel “querydriven ” update policy, which reduces the update cost of conventional policies significantly. 1.
Solving Integer and Disjunctive Programs by LiftandProject (Extended Abstract)
, 1998
"... ) Sebasti'an Ceria 1 and G'abor Pataki 2 ? 1 Graduate School of Business and Computational Optimization Research Center Columbia University, New York, NY 10027 sebas@cumparsita.gsb.columbia.edu, http://www.columbia.edu/~sc244 2 Department of Industrial Engineering and Operations Res ..."
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Cited by 8 (1 self)
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) Sebasti'an Ceria 1 and G'abor Pataki 2 ? 1 Graduate School of Business and Computational Optimization Research Center Columbia University, New York, NY 10027 sebas@cumparsita.gsb.columbia.edu, http://www.columbia.edu/~sc244 2 Department of Industrial Engineering and Operations Research and Computational Optimization Research Center Columbia University, New York, NY 10027 gabor@ieor.columbia.edu, http://www.ieor.columbia.edu/~gabor Abstract. We extend the theoretical foundations of the branchandcut method using liftandproject cuts for a broader class of disjunctive constraints, and also present a new, substantially improved disjunctive cut generator. Employed together with an efficient commercial MIP solver, our code is a robust, general purpose method for solving mixed integer programs. We present extensive computational experience with the most difficult problems in the MIPLIB library. 1 Introduction Disjunctive programming is optimization over a finite union of convex ...
Systematic Modeling of DiscreteContinuous Optimization Models through Generalized Disjunctive Programming
"... This article is dedicated to the memory of Professor Neil Amundsen, who pioneered the application of mathematical modeling and analysis in chemical engineering. Discretecontinuous optimization problems in process systems engineering are commonly modeled in algebraic form as mixedinteger linear or ..."
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Cited by 7 (5 self)
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This article is dedicated to the memory of Professor Neil Amundsen, who pioneered the application of mathematical modeling and analysis in chemical engineering. Discretecontinuous optimization problems in process systems engineering are commonly modeled in algebraic form as mixedinteger linear or nonlinear programming models. Since these models can often be formulated in different ways, there is a need for a systematic modeling framework that provides a fundamental understanding on the nature of these models, particularly their continuous relaxations. This paper describes a modeling framework, Generalized Disjunctive Programming (GDP), which represents problems in terms of Boolean and continuous variables, allowing the representation of constraints as algebraic equations, disjunctions and logic propositions. We provide an overview of major research results that have emerged in this area. Basic concepts are emphasized as well as major classes of formulations that can be derived. These are illustrated with a number of examples in the area of process systems engineering. As will be shown, GDP provides a structured way for systematically deriving mixedinteger optimization models that exhibit strong continuous relaxations. 1.