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Polyhedral approaches to machine scheduling
, 1996
"... We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition ..."
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Cited by 37 (8 self)
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We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition to those just required for the validity of the initial formulation, in order to obtain useful lower bounds and structural insights. We review formulations based on time–indexed variables; on linear ordering, start time and completion time variables; on assignment and positional date variables; and on traveling salesman variables. We point out relationship between various models, and provide a number of new results, as well as simplified new proofs of known results. In particular, we emphasize the important role that supermodular polyhedra and greedy algorithms play in many formulations and we analyze the strength of the lower and upper bounds obtained from different formulations and relaxations. We discuss separation algorithms for several classes of inequalities, and their potential applicability in generating cutting planes for the practical solution of such scheduling problems. We also review some recent results on approximation algorithms based on some of these formulations.
Convexity Recognition of the Union of Polyhedra
, 2000
"... In this paper we consider the following basic problem in polyhedral computation: Given two polyhedra in R d , P and Q, decide whether their union is convex, and, if so, compute it. We consider the three natural specializations of the problem: (1) when the polyhedra are given by halfspaces (Hpolyh ..."
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Cited by 20 (6 self)
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In this paper we consider the following basic problem in polyhedral computation: Given two polyhedra in R d , P and Q, decide whether their union is convex, and, if so, compute it. We consider the three natural specializations of the problem: (1) when the polyhedra are given by halfspaces (Hpolyhedra) (2) when they are given by vertices and extreme rays (Vpolyhedra) (3) when both H and Vpolyhedral representations are available. Both the bounded (polytopes) and the unbounded case are considered. We show that the first two problems are polynomially solvable, and that the third problem is stronglypolynomially solvable.
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.
Extended Convex Hull
, 2000
"... In this paper we address the problem of computing a minimal Hrepresentation of the convex hull of the union of k Hpolytopes in R^d. Our method applies the reverse search algorithm to a shelling ordering of the facets of the convex hull. Efficient wrapping is done by projecting the polytopes onto t ..."
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Cited by 7 (1 self)
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In this paper we address the problem of computing a minimal Hrepresentation of the convex hull of the union of k Hpolytopes in R^d. Our method applies the reverse search algorithm to a shelling ordering of the facets of the convex hull. Efficient wrapping is done by projecting the polytopes onto the twodimensional space and solving a linear program. The resulting algorithm is polynomial in the sizes of input and output under the general position assumption.
On the hardness of minkowski addition and related operations
 In Symposium on Computational Geometry
, 2007
"... For polytopes P,Q ⊂ Rd we consider the intersection P ∩Q; the convex hull of the union CH(P ∪Q); and the Minkowski sum P +Q. We prove that given rational Hpolytopes P1,P2,Q it is impossible to verify in polynomial time whether Q = P1 + P2, unless P = NP. In particular, this shows that there is no o ..."
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Cited by 4 (3 self)
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For polytopes P,Q ⊂ Rd we consider the intersection P ∩Q; the convex hull of the union CH(P ∪Q); and the Minkowski sum P +Q. We prove that given rational Hpolytopes P1,P2,Q it is impossible to verify in polynomial time whether Q = P1 + P2, unless P = NP. In particular, this shows that there is no output sensitive polynomial algorithm to compute the facets of the Minkowski sum of two arbitrary Hpolytopes even if we consider only rational polytopes. Since the convex hull of the union and the intersection of two polytopes relate naturally to the Minkowski sum via the Cayley trick and polarity, similar hardness results follow for these operations as well.
On the hardness of computing intersection, union and minkowski sum of polytopes
 DISCRETE & COMPUTATIONAL GEOMETRY
"... For polytopes P1, P2 ⊂ R d we consider the intersection P1 ∩ P2, the convex hull of the union CH(P1 ∪ P2), and the Minkowski sum P1 + P2. For Minkowski sum we prove that enumerating the facets of P1+P2 is NPhard if P1 and P2 are specified by facets, or if P1 is specified by vertices and P2 is a poly ..."
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Cited by 4 (1 self)
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For polytopes P1, P2 ⊂ R d we consider the intersection P1 ∩ P2, the convex hull of the union CH(P1 ∪ P2), and the Minkowski sum P1 + P2. For Minkowski sum we prove that enumerating the facets of P1+P2 is NPhard if P1 and P2 are specified by facets, or if P1 is specified by vertices and P2 is a polyhedral cone specified by facets. For intersection we prove that computing the facets or the vertices of the intersection of two polytopes is NPhard if one of them is given by vertices and the other by facets. Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NPhard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NPcomplete.
Solving Factorable Programs with Applications to Cluster Analysis,
, 2005
"... Despite recent advances in optimization research and computing technology, deriving global optimal solutions to nonconvex optimization problems remains a daunting task. Existing approaches for solving such formidable problems are typically heuristic in nature, often leading to significantly subopti ..."
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Despite recent advances in optimization research and computing technology, deriving global optimal solutions to nonconvex optimization problems remains a daunting task. Existing approaches for solving such formidable problems are typically heuristic in nature, often leading to significantly suboptimal solutions. This motivates the need to develop a framework for optimally solving a broad class of nonconvex programming problems, which yet retains sufficient flexibility to exploit inherent special structures. Toward this end, we focus in this dissertation on a variety of applications that occur in practice as instances of polynomial programming problems or more general nonconvex factorable programs, and we employ a central theme based on the ReformulationLinearization Technique (RLT) to design theoretically convergent and practically effective and robust solution methodologies. We begin our discussion in this dissertation by providing a basis for developing efficient solution methodologies for solving the class of nonconvex factorable programming problems. Recognizing the ability of the RLT to solve polynomial programs to (global) optimality, the basic idea is to solve the given nonconvex program
Relaxations for twolevel multiitem lotsizing problem
, 2012
"... We consider several variants of the twolevel lotsizing problem with one item at the upper level facing dependent demand, and multiple items or clients at the lower level, facing independent demands. We first show that under a natural cost assumption, it is sufficient to optimize over a stockdomin ..."
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We consider several variants of the twolevel lotsizing problem with one item at the upper level facing dependent demand, and multiple items or clients at the lower level, facing independent demands. We first show that under a natural cost assumption, it is sufficient to optimize over a stockdominant relaxation. We further study the polyhedral structure of a strong relaxation of this problem involving only initial inventory variables and setup variables. We consider several variants: uncapacitated at both levels with or without startup costs, uncapacitated at the upper level and constant capacity at the lower level, constant capacity at both levels. We finally demonstrate how the strong formulations described improve our ability to solve instances with up to several dozens of periods and a few hundred products.
Modeling of discrete/continuous optimization problems: characterization and formulation of disjunctions and their relaxations
"... 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 prop ..."
Abstract
 Add to MetaCart
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.