Results 1  10
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88
Branchandprice: Column generation for solving huge integer programs
 Oper. Res
, 1998
"... We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branchandbound tree. We present classes of models for which th ..."
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Cited by 208 (8 self)
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We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branchandbound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. Wethen discuss computational issues and implementation of column generation, branchandbound algorithms, including special branching rules and e cient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality. 1
Selected topics in column generation
 Operations Research
, 2002
"... DantzigWolfe decomposition and column generation, devised for linear programs, is a success story in large scale integer programming. We outline and relate the approaches, and survey mainly recent contributions, not found in textbooks, yet. We emphasize on the growing understanding of the dual poin ..."
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Cited by 71 (5 self)
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DantzigWolfe decomposition and column generation, devised for linear programs, is a success story in large scale integer programming. We outline and relate the approaches, and survey mainly recent contributions, not found in textbooks, yet. We emphasize on the growing understanding of the dual point of view, which brought considerable progress to the column generation theory and practice. It stimulated careful initializations, sophisticated solution techniques for restricted master problem and subproblem, as well as better overall performance. Thus, the dual perspective is an ever recurring concept in our "selected topics."
Convex Nondifferentiable Optimization: A Survey Focussed On The Analytic Center Cutting Plane Method.
, 1999
"... We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct pr ..."
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Cited by 51 (2 self)
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We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.
The convex hull of two core capacitated network design problems
 MATHEMATICAL PROGRAMMING
, 1993
"... The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed pointtopoint demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs ..."
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Cited by 42 (0 self)
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The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed pointtopoint demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost. This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.
Approximate Dynamic Programming For Sensor Management
, 1997
"... This paper studies the problem of dynamic scheduling of multimode sensor resources for the problem of classification of multiple unknown objects. Because of the uncertain nature of the object types, the problem is formulated as a partially observed Markov decision problem with a large state space. ..."
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Cited by 39 (0 self)
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This paper studies the problem of dynamic scheduling of multimode sensor resources for the problem of classification of multiple unknown objects. Because of the uncertain nature of the object types, the problem is formulated as a partially observed Markov decision problem with a large state space. The paper describes a hierarchical algorithm approach for e#cient solution of sensor scheduling problems with large numbers of objects, based on combination of stochastic dynamic programming and nondi#erentiable optimization techniques. The algorithm is illustrated with an application involving classification of 10,000 unknown objects. 1 Introduction Many modern avionics systems include multiple sensors as well as individual sensors capable of focusing on different objects with di#erent modes. In order to achieve an accurate possible representation of all objects of interest, it is important to coordinate the allocation and scheduling of the di#erent sensors and sensor modes across the di#...
Optimal ShortTerm Scheduling of LargeScale Power Systems
, 1983
"... This paper is concerned with the longstanding problem of optimal unit commitment in an electric power system. We follow the traditional formulation of this problem which gives rise to a largescale, dynamic, mixedinteger programming problem. We describe a solution methodology based on duality, Lagr ..."
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Cited by 38 (0 self)
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This paper is concerned with the longstanding problem of optimal unit commitment in an electric power system. We follow the traditional formulation of this problem which gives rise to a largescale, dynamic, mixedinteger programming problem. We describe a solution methodology based on duality, Lagrangian relaxation and nondifferentiable optimization that has two unique features. First, computational requirements typically grow only linearly witb the number of generating units. Second, the duality gap decreases in relative terms as the number of units increases, and as a result our algorithm tends to actually perform better for problems of large size. This allows for the first time consistently reliable solution of large practical problems involving several hundreds of units within realistic time constraints. Aside from the unit commilment problem. this methodology is applicable to a broad class of largescale dpamic scheduling and resource allocation problems involving integer variables.
The Theory of Discrete Lagrange Multipliers for Nonlinear Discrete Optimization
 Principles and Practice of Constraint Programming
, 1999
"... In this paper we present a Lagrangemultiplier formulation of discrete constrained optimization problems, the associated discretespace firstorder necessary and sufficient conditions for saddle points, and an efficient firstorder search procedure that looks for saddle points in discrete space. Our ..."
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Cited by 38 (21 self)
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In this paper we present a Lagrangemultiplier formulation of discrete constrained optimization problems, the associated discretespace firstorder necessary and sufficient conditions for saddle points, and an efficient firstorder search procedure that looks for saddle points in discrete space. Our new theory provides a strong mathematical foundation for solving general nonlinear discrete optimization problems. Specifically, we propose a new vectorbased definition of descent directions in discrete space and show that the new definition does not obey the rules of calculus in continuous space. Starting from the concept of saddle points and using only vector calculus, we then prove the discretespace firstorder necessary and sufficient conditions for saddle points. Using welldefined transformations on the constraint functions, we further prove that the set of discretespace saddle points is the same as the set of constrained local minima, leading to the firstorder necessary and sufficient conditions for constrained local minima. Based on the firstorder conditions, we propose a localsearch method to look for saddle points that satisfy the firstorder conditions.
On Integrating Constraint Propagation and Linear Programming for Combinatorial Optimization
, 2000
"... Integer programming and constraint (logic) programming are two traditional techniques for solving combinatorial optimization problems; the former based on linear programming relaxations and the latter on constraint propagation. Attempts to combine them have mainly focused on incorporating either ..."
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Cited by 36 (10 self)
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Integer programming and constraint (logic) programming are two traditional techniques for solving combinatorial optimization problems; the former based on linear programming relaxations and the latter on constraint propagation. Attempts to combine them have mainly focused on incorporating either technique into the framework of the other traditional models have been left intact. We argue that a rethinking of our modeling traditions is necessary to achieve the greatest benet of such an integration. We propose a declarative modeling framework in which the structure of the constraints indicates how LP and CP can interact to solve the problem. 1 Introduction Linear programming (LP) and constraint propagation (CP) are two complementary techniques with potential for integration to benet the solution of combinatorial optimization problems. Integer programming (IP) has been successfully applied to a range of problems, such as capital budgeting, bin packing and traveling salesman pr...
Semidefinite relaxations and Lagrangian duality with application to combinatorial optimization
, 1999
"... We show that it is fruitful to dualize the integrality constraints in a combinatorial optimization problem. First, this reproduces the known SDP relaxations of the maxcut and maxstable problems. Then we apply the approach to general combinatorial problems. We show that the resulting duality gap is ..."
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Cited by 34 (0 self)
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We show that it is fruitful to dualize the integrality constraints in a combinatorial optimization problem. First, this reproduces the known SDP relaxations of the maxcut and maxstable problems. Then we apply the approach to general combinatorial problems. We show that the resulting duality gap is smaller than with the classical Lagrangian relaxation; we also show that linear constraints need a special treatment.
Scheduling Of Manufacturing Systems Using The Lagrangian Relaxation Technique
 IEEE Transactions on Automatic Control
, 1993
"... Scheduling is one of the most basic but the most difficult problems encountered in the manufacturing industry. Generally, some degree of timeconsuming and impractical enumeration is required to obtain optimal solutions. Industry has thus relied on a combination of heuristics and simulation to solve ..."
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Cited by 25 (9 self)
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Scheduling is one of the most basic but the most difficult problems encountered in the manufacturing industry. Generally, some degree of timeconsuming and impractical enumeration is required to obtain optimal solutions. Industry has thus relied on a combination of heuristics and simulation to solve the problem, resulting in unreliable and often infeasible schedules. Yet, there is a great need for an improvement in scheduling operations in complex and turbulent manufacturing environments. The logical strategy is to find scheduling methods which consistently generate good schedules efficiently. However, it is often difficult to measure the quality of a schedule without knowing the optimum. In this paper, the practical scheduling of three manufacturing environments are examined in the increasing order of complexity. The first problem considers scheduling singleoperation jobs on parallel, identical machines; the second one is concerned with scheduling multipleoperation jobs with simple ...