George Nemhauser | Georgia Institute of Technology (original) (raw)
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Papers by George Nemhauser
Operations Research Letters, Jan 1, 1998
Mathematical Programming, 1993
2000 Winter Simulation Conference Proceedings (Cat. No.00CH37165), 2000
Transportation Science, 2003
... Jay M. Rosenberger, Ellis L. Johnson, George L. Nemhauser American Airlines Operations Resear... more ... Jay M. Rosenberger, Ellis L. Johnson, George L. Nemhauser American Airlines Operations Research and Decision Support Department, MD 5358, PO Box 619616, Dallas/Fort Worth Airport, Texas 75261-9616 School of Industrial and Systems Engineering, Georgia Institute of ...
Operations Research, 1995
Airline crew scheduling is concerned with finding a minimum cost assignmentof flight crews to a g... more Airline crew scheduling is concerned with finding a minimum cost assignmentof flight crews to a given flight schedule while satisfying restrictions dictatedby collective bargaining agreements and the Federal Aviation Administration.Traditionally, the problem has been modeled as a set partitioning problem.In this paper, we present a new model based on breaking the decision processinto two stages. In the first stage we
Nemhauser/Integer and Combinatorial Optimization, 1988
ABSTRACT The structure invoked in this chapter is that the problems have only one constraint othe... more ABSTRACT The structure invoked in this chapter is that the problems have only one constraint other than bounds and integrality on the variables. It considers the integer knapsack problem, the group problem, and the 0-1 knapsack problem. The chapter introduces the uncapacitated lot-size problem (ULS) using the formulation, and then reformulates it as an uncapacitated facility location problem. The use of basis reduction in lattices is new to integer programming. To indicate its potential, the chapter outlines two applications. The first is a simple heuristic algorithm to find a feasible solution to a 0-1 equality knapsack constraint. The second is an algorithm for integer programming that is polynomial for fixed n. Although this is an important theoretical result, the algorithm is not practical. The result has, however, motivated the application of basis reduction techniques to a variety of problems.
Nemhauser/Integer and Combinatorial Optimization, 1988
ABSTRACT The theme of this chapter is to use structure to determine strong valid inequalities for... more ABSTRACT The theme of this chapter is to use structure to determine strong valid inequalities for the constraint sets of some N-P hard integer programming problems. The determination of families of strong valid inequalities is more of an art than a formal methodology. Thus the presentation will largely be a series of examples that convey the basic ideas. The mathematics enters in proving that classes of inequalities, which are often easily shown to be valid, are indeed strong in the sense that they define facets or faces of reasonable dimension. A related mathematical problem, is to prove that a given family of inequalities represents all of the facets of the convex hull.
Nemhauser/Integer and Combinatorial Optimization, 1988
Nemhauser/Integer and Combinatorial Optimization, 1988
ABSTRACT This chapter presents some polynomial-time algorithms for linear programming and discuss... more ABSTRACT This chapter presents some polynomial-time algorithms for linear programming and discusses their consequences in combinatorial optimization. It talks about the ellipsoid algorithm, which was acclaimed on the front pages of newspapers throughout the world when it appeared in 1979. Although the algorithm turned out to be computationally impractical, it yielded important theoretical results. It was the first polynomial-time algorithm for linear programming. The ellipsoid algorithm is a tool for proving that certain combinatorial optimization problems can be solved in polynomial time. The chapter also presents a version of a polynomial-time projective algorithm for linear programming. It shows the significance of projections in solving linear programs.
Nemhauser/Integer and Combinatorial Optimization, 1988
ABSTRACT Matroids and submodular functions are the foundations for some combinatorial optimizatio... more ABSTRACT Matroids and submodular functions are the foundations for some combinatorial optimization problems that generalize both network flow problems and the spanning tree problem. Matroids can be viewed as prototypes of independence systems and 0-1 integer programs with “nice” properties that can be used to obtain efficient algorithms for the corresponding optimization problems. Matroids were originally developed from matrices to generalize the properties of linear independence and bases in a vector space. This generalization has yielded several classes of matroids that include: matric matroids, graphic matroids and partition matroids. This chapter establishes the equivalence between a matroid M = (N, 9) and a submodular rank function r on TV. It also introduces and develops some elementary matroid properties. The matroid optimization problem is also considered. The chapter further studies efficient algorithms for the 2-matroid intersection problem.
Nemhauser/Integer and Combinatorial Optimization, 1988
ABSTRACT This chapter discusses representation of an integer program by a linear program that has... more ABSTRACT This chapter discusses representation of an integer program by a linear program that has the same optimal solution. It shows the relationship between the valid inequalities used by Gomory and the rounding procedure. The chapter develops a procedure for generating valid inequalities for T. Note that the C-G procedure does not work when there are continuous variables. In particular, we cannot round down the right-hand side of an inequality to its integer part when all of the coefficients on the left-hand side are integers. The chapter extends the development of superadditive valid inequalities to mixed-integer constraint sets. Superadditive inequalities for 0-1 problems were studied by Wolsey (1977), and those for multiple right-hand side problems were studied by Johnson (1981b).
Handbooks in Operations Research and Management Science, 2005
We review and describe several results regarding integer programming problems in fixed dimension.... more We review and describe several results regarding integer programming problems in fixed dimension. First, we describe various lattice basis reduction algorithms that are used as auxiliary algorithms when solving integer feasibility and optimization problems. Next, we review three algorithms for solving the integer feasibility problem. These algorithms are based on the idea of branching on lattice hyperplanes, and their running
Operations Research Letters, 1994
ABSTRACT
Lecture Notes in Computer Science, 2001
ABSTRACT
Progress in Material Handling and Logistics, 1991
Recent geopolitical changes, the continuing development of truly global corporations, and theincr... more Recent geopolitical changes, the continuing development of truly global corporations, and theincreasing importance of make-to-order manufacturing and inventory reductions in distribution systems haveplaced further emphasis on the growing role of logistics in the current business environment. One of the needsmost often identified by companies is a tool to rapidly evaluate and design different logistics philosophies andconfigurations. This tool must be
Lecture Notes in Computer Science, 2003
We present a finitely convergent cutting plane algorithm for 0-1 mixed integer programming. The a... more We present a finitely convergent cutting plane algorithm for 0-1 mixed integer programming. The algorithm is a hybrid between a strong cutting plane and a Gomory-type algorithm that generates violated facet-defining inequalities of a relaxation of the simplex tableau and uses them as cuts for the original problem. We show that the cuts can be computed in polynomial time and
Operations Research Letters, Jan 1, 1998
Mathematical Programming, 1993
2000 Winter Simulation Conference Proceedings (Cat. No.00CH37165), 2000
Transportation Science, 2003
... Jay M. Rosenberger, Ellis L. Johnson, George L. Nemhauser American Airlines Operations Resear... more ... Jay M. Rosenberger, Ellis L. Johnson, George L. Nemhauser American Airlines Operations Research and Decision Support Department, MD 5358, PO Box 619616, Dallas/Fort Worth Airport, Texas 75261-9616 School of Industrial and Systems Engineering, Georgia Institute of ...
Operations Research, 1995
Airline crew scheduling is concerned with finding a minimum cost assignmentof flight crews to a g... more Airline crew scheduling is concerned with finding a minimum cost assignmentof flight crews to a given flight schedule while satisfying restrictions dictatedby collective bargaining agreements and the Federal Aviation Administration.Traditionally, the problem has been modeled as a set partitioning problem.In this paper, we present a new model based on breaking the decision processinto two stages. In the first stage we
Nemhauser/Integer and Combinatorial Optimization, 1988
ABSTRACT The structure invoked in this chapter is that the problems have only one constraint othe... more ABSTRACT The structure invoked in this chapter is that the problems have only one constraint other than bounds and integrality on the variables. It considers the integer knapsack problem, the group problem, and the 0-1 knapsack problem. The chapter introduces the uncapacitated lot-size problem (ULS) using the formulation, and then reformulates it as an uncapacitated facility location problem. The use of basis reduction in lattices is new to integer programming. To indicate its potential, the chapter outlines two applications. The first is a simple heuristic algorithm to find a feasible solution to a 0-1 equality knapsack constraint. The second is an algorithm for integer programming that is polynomial for fixed n. Although this is an important theoretical result, the algorithm is not practical. The result has, however, motivated the application of basis reduction techniques to a variety of problems.
Nemhauser/Integer and Combinatorial Optimization, 1988
ABSTRACT The theme of this chapter is to use structure to determine strong valid inequalities for... more ABSTRACT The theme of this chapter is to use structure to determine strong valid inequalities for the constraint sets of some N-P hard integer programming problems. The determination of families of strong valid inequalities is more of an art than a formal methodology. Thus the presentation will largely be a series of examples that convey the basic ideas. The mathematics enters in proving that classes of inequalities, which are often easily shown to be valid, are indeed strong in the sense that they define facets or faces of reasonable dimension. A related mathematical problem, is to prove that a given family of inequalities represents all of the facets of the convex hull.
Nemhauser/Integer and Combinatorial Optimization, 1988
Nemhauser/Integer and Combinatorial Optimization, 1988
ABSTRACT This chapter presents some polynomial-time algorithms for linear programming and discuss... more ABSTRACT This chapter presents some polynomial-time algorithms for linear programming and discusses their consequences in combinatorial optimization. It talks about the ellipsoid algorithm, which was acclaimed on the front pages of newspapers throughout the world when it appeared in 1979. Although the algorithm turned out to be computationally impractical, it yielded important theoretical results. It was the first polynomial-time algorithm for linear programming. The ellipsoid algorithm is a tool for proving that certain combinatorial optimization problems can be solved in polynomial time. The chapter also presents a version of a polynomial-time projective algorithm for linear programming. It shows the significance of projections in solving linear programs.
Nemhauser/Integer and Combinatorial Optimization, 1988
ABSTRACT Matroids and submodular functions are the foundations for some combinatorial optimizatio... more ABSTRACT Matroids and submodular functions are the foundations for some combinatorial optimization problems that generalize both network flow problems and the spanning tree problem. Matroids can be viewed as prototypes of independence systems and 0-1 integer programs with “nice” properties that can be used to obtain efficient algorithms for the corresponding optimization problems. Matroids were originally developed from matrices to generalize the properties of linear independence and bases in a vector space. This generalization has yielded several classes of matroids that include: matric matroids, graphic matroids and partition matroids. This chapter establishes the equivalence between a matroid M = (N, 9) and a submodular rank function r on TV. It also introduces and develops some elementary matroid properties. The matroid optimization problem is also considered. The chapter further studies efficient algorithms for the 2-matroid intersection problem.
Nemhauser/Integer and Combinatorial Optimization, 1988
ABSTRACT This chapter discusses representation of an integer program by a linear program that has... more ABSTRACT This chapter discusses representation of an integer program by a linear program that has the same optimal solution. It shows the relationship between the valid inequalities used by Gomory and the rounding procedure. The chapter develops a procedure for generating valid inequalities for T. Note that the C-G procedure does not work when there are continuous variables. In particular, we cannot round down the right-hand side of an inequality to its integer part when all of the coefficients on the left-hand side are integers. The chapter extends the development of superadditive valid inequalities to mixed-integer constraint sets. Superadditive inequalities for 0-1 problems were studied by Wolsey (1977), and those for multiple right-hand side problems were studied by Johnson (1981b).
Handbooks in Operations Research and Management Science, 2005
We review and describe several results regarding integer programming problems in fixed dimension.... more We review and describe several results regarding integer programming problems in fixed dimension. First, we describe various lattice basis reduction algorithms that are used as auxiliary algorithms when solving integer feasibility and optimization problems. Next, we review three algorithms for solving the integer feasibility problem. These algorithms are based on the idea of branching on lattice hyperplanes, and their running
Operations Research Letters, 1994
ABSTRACT
Lecture Notes in Computer Science, 2001
ABSTRACT
Progress in Material Handling and Logistics, 1991
Recent geopolitical changes, the continuing development of truly global corporations, and theincr... more Recent geopolitical changes, the continuing development of truly global corporations, and theincreasing importance of make-to-order manufacturing and inventory reductions in distribution systems haveplaced further emphasis on the growing role of logistics in the current business environment. One of the needsmost often identified by companies is a tool to rapidly evaluate and design different logistics philosophies andconfigurations. This tool must be
Lecture Notes in Computer Science, 2003
We present a finitely convergent cutting plane algorithm for 0-1 mixed integer programming. The a... more We present a finitely convergent cutting plane algorithm for 0-1 mixed integer programming. The algorithm is a hybrid between a strong cutting plane and a Gomory-type algorithm that generates violated facet-defining inequalities of a relaxation of the simplex tableau and uses them as cuts for the original problem. We show that the cuts can be computed in polynomial time and