Discrete optimization:
Gespeichert in:
| Format: | Elektronisch E-Book |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Amsterdam
Elsevier
2005
|
| Ausgabe: | 1st ed |
| Schriftenreihe: | Handbooks in operations research and management science
v. 12 |
| Schlagworte: | |
| Online-Zugang: | FAW01 Volltext |
| Beschreibung: | Includes bibliographical references and index The chapters of this Handbook volume covers nine main topics that are representative of recent theoretical and algorithmic developments in the field. In addition to the nine papers that present the state of the art, there is an article on the early history of the field. The handbook will be a useful reference to experts in the field as well as students and others who want to learn about discrete optimization. All of the chapters in this handbook are written by authors who have made significant original contributions to their topics. Herewith a brief introduction to the chapters of the handbook. "On the history of combinatorial optimization (until 1960)" goes back to work of Monge in the 18th century on the assignment problem and presents six problem areas: assignment, transportation, maximum flow, shortest tree, shortest path and traveling salesman. The branch-and-cut algorithm of integer programming is the computational workhorse of discrete optimization. It provides the tools that have been implemented in commercial software such as CPLEX and Xpress MP that make it possible to solve practical problems in supply chain, manufacturing, telecommunications and many other areas. "Computational integer programming and cutting planes" presents the key ingredients of these algorithms. Although branch-and-cut based on linear programming relaxation is the most widely used integer programming algorithm, other approaches are needed to solve instances for which branch-and-cut performs poorly and to understand better the structure of integral polyhedra. The next three chapters discuss alternative approaches. "The structure of group relaxations" studies a family of polyhedra obtained by dropping certain nonnegativity restrictions on integer programming problems. Although integer programming is NP-hard in general, it is polynomially solvable in fixed dimension. "Integer programming, lattices, and results in fixed dimension" presents results in this area including algorithms that use reduced bases of integer lattices that are capable of solving certain classes of integer programs that defy solution by branch-and-cut. Relaxation or dual methods, such as cutting plane algorithms, progressively remove infeasibility while maintaining optimality to the relaxed problem. Such algorithms have the disadvantage of possibly obtaining feasibility only when the algorithm terminates. Primal methods for integer programs, which move from a feasible solution to a better feasible solution, were studied in the 1960's but did not appear to be competitive with dual methods. However, recent development in primal methods presented in "Primal integer programming" indicate that this approach is not just interesting theoretically but may have practical implications as well. The study of matrices that yield integral polyhedra has a long tradition in integer programming. A major breakthrough occurred in the 1990's with the development of polyhedral and structural results and recognition algorithms for balanced matrices. "Balanced matrices" is a tutorial on the subject. Submodular function minimization generalizes some linear combinatorial optimization problems such as minimum cut and is one of the fundamental problems of the field that is solvable in polynomial time. "Submodular function minimization" presents the theory and algorithms of this subject. In the search for tighter relaxations of combinatorial optimization problems, semidefinite programming provides a generalization of linear programming that can give better approximations and is still polynomially solvable. This subject is discussed in "Semidefinite programming and integer programming". Many real world problems have uncertain data that is known only probabilistically. Stochastic programming treats this topic, but until recently it was limited, for computational reasons, to stochastic linear programs. Stochastic integer programming is now a high profile research area and recent developments are presented in "Algorithms for stochastic mixed-integer programming models". Resource constrained scheduling is an example of a class of combinatorial optimization problems that is not naturally formulated with linear constraints so that linear programming based methods do not work well. "Constraint programming" presents an alternative enumerative approach that is complementary to branch-and-cut. Constraint programming, primarily designed for feasibility problems, does not use a relaxation to obtain bounds. Instead nodes of the search tree are pruned by constraint propagation, which tightens bounds on variables until their values are fixed or their domains are shown to be empty |
| Beschreibung: | xi, 607 pages |
| ISBN: | 0444515070 9780444515070 0080459218 9780080459219 |
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| 500 | |a It provides the tools that have been implemented in commercial software such as CPLEX and Xpress MP that make it possible to solve practical problems in supply chain, manufacturing, telecommunications and many other areas. "Computational integer programming and cutting planes" presents the key ingredients of these algorithms. Although branch-and-cut based on linear programming relaxation is the most widely used integer programming algorithm, other approaches are needed to solve instances for which branch-and-cut performs poorly and to understand better the structure of integral polyhedra. The next three chapters discuss alternative approaches. "The structure of group relaxations" studies a family of polyhedra obtained by dropping certain nonnegativity restrictions on integer programming problems. Although integer programming is NP-hard in general, it is polynomially solvable in fixed dimension. | ||
| 500 | |a "Integer programming, lattices, and results in fixed dimension" presents results in this area including algorithms that use reduced bases of integer lattices that are capable of solving certain classes of integer programs that defy solution by branch-and-cut. Relaxation or dual methods, such as cutting plane algorithms, progressively remove infeasibility while maintaining optimality to the relaxed problem. Such algorithms have the disadvantage of possibly obtaining feasibility only when the algorithm terminates. Primal methods for integer programs, which move from a feasible solution to a better feasible solution, were studied in the 1960's but did not appear to be competitive with dual methods. However, recent development in primal methods presented in "Primal integer programming" indicate that this approach is not just interesting theoretically but may have practical implications as well. | ||
| 500 | |a The study of matrices that yield integral polyhedra has a long tradition in integer programming. A major breakthrough occurred in the 1990's with the development of polyhedral and structural results and recognition algorithms for balanced matrices. "Balanced matrices" is a tutorial on the subject. Submodular function minimization generalizes some linear combinatorial optimization problems such as minimum cut and is one of the fundamental problems of the field that is solvable in polynomial time. "Submodular function minimization" presents the theory and algorithms of this subject. In the search for tighter relaxations of combinatorial optimization problems, semidefinite programming provides a generalization of linear programming that can give better approximations and is still polynomially solvable. This subject is discussed in "Semidefinite programming and integer programming". Many real world problems have uncertain data that is known only probabilistically. | ||
| 500 | |a Stochastic programming treats this topic, but until recently it was limited, for computational reasons, to stochastic linear programs. Stochastic integer programming is now a high profile research area and recent developments are presented in "Algorithms for stochastic mixed-integer programming models". Resource constrained scheduling is an example of a class of combinatorial optimization problems that is not naturally formulated with linear constraints so that linear programming based methods do not work well. "Constraint programming" presents an alternative enumerative approach that is complementary to branch-and-cut. Constraint programming, primarily designed for feasibility problems, does not use a relaxation to obtain bounds. Instead nodes of the search tree are pruned by constraint propagation, which tightens bounds on variables until their values are fixed or their domains are shown to be empty | ||
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| spelling | Discrete optimization edited by K. Aardal, G.L. Nemhauser and R. Weismantel 1st ed Amsterdam Elsevier 2005 xi, 607 pages txt rdacontent c rdamedia cr rdacarrier Handbooks in operations research and management science v. 12 Includes bibliographical references and index The chapters of this Handbook volume covers nine main topics that are representative of recent theoretical and algorithmic developments in the field. In addition to the nine papers that present the state of the art, there is an article on the early history of the field. The handbook will be a useful reference to experts in the field as well as students and others who want to learn about discrete optimization. All of the chapters in this handbook are written by authors who have made significant original contributions to their topics. Herewith a brief introduction to the chapters of the handbook. "On the history of combinatorial optimization (until 1960)" goes back to work of Monge in the 18th century on the assignment problem and presents six problem areas: assignment, transportation, maximum flow, shortest tree, shortest path and traveling salesman. The branch-and-cut algorithm of integer programming is the computational workhorse of discrete optimization. It provides the tools that have been implemented in commercial software such as CPLEX and Xpress MP that make it possible to solve practical problems in supply chain, manufacturing, telecommunications and many other areas. "Computational integer programming and cutting planes" presents the key ingredients of these algorithms. Although branch-and-cut based on linear programming relaxation is the most widely used integer programming algorithm, other approaches are needed to solve instances for which branch-and-cut performs poorly and to understand better the structure of integral polyhedra. The next three chapters discuss alternative approaches. "The structure of group relaxations" studies a family of polyhedra obtained by dropping certain nonnegativity restrictions on integer programming problems. Although integer programming is NP-hard in general, it is polynomially solvable in fixed dimension. "Integer programming, lattices, and results in fixed dimension" presents results in this area including algorithms that use reduced bases of integer lattices that are capable of solving certain classes of integer programs that defy solution by branch-and-cut. Relaxation or dual methods, such as cutting plane algorithms, progressively remove infeasibility while maintaining optimality to the relaxed problem. Such algorithms have the disadvantage of possibly obtaining feasibility only when the algorithm terminates. Primal methods for integer programs, which move from a feasible solution to a better feasible solution, were studied in the 1960's but did not appear to be competitive with dual methods. However, recent development in primal methods presented in "Primal integer programming" indicate that this approach is not just interesting theoretically but may have practical implications as well. The study of matrices that yield integral polyhedra has a long tradition in integer programming. A major breakthrough occurred in the 1990's with the development of polyhedral and structural results and recognition algorithms for balanced matrices. "Balanced matrices" is a tutorial on the subject. Submodular function minimization generalizes some linear combinatorial optimization problems such as minimum cut and is one of the fundamental problems of the field that is solvable in polynomial time. "Submodular function minimization" presents the theory and algorithms of this subject. In the search for tighter relaxations of combinatorial optimization problems, semidefinite programming provides a generalization of linear programming that can give better approximations and is still polynomially solvable. This subject is discussed in "Semidefinite programming and integer programming". Many real world problems have uncertain data that is known only probabilistically. Stochastic programming treats this topic, but until recently it was limited, for computational reasons, to stochastic linear programs. Stochastic integer programming is now a high profile research area and recent developments are presented in "Algorithms for stochastic mixed-integer programming models". Resource constrained scheduling is an example of a class of combinatorial optimization problems that is not naturally formulated with linear constraints so that linear programming based methods do not work well. "Constraint programming" presents an alternative enumerative approach that is complementary to branch-and-cut. Constraint programming, primarily designed for feasibility problems, does not use a relaxation to obtain bounds. Instead nodes of the search tree are pruned by constraint propagation, which tightens bounds on variables until their values are fixed or their domains are shown to be empty Optimisation mathématique Programmation en nombres entiers MATHEMATICS / Optimization bisacsh Integer programming fast Mathematical optimization fast Diskrete Optimierung swd Mathematical optimization Integer programming Diskrete Optimierung (DE-588)4150179-2 gnd rswk-swf Diskrete Optimierung (DE-588)4150179-2 s 1\p DE-604 Aardal, K. Sonstige oth Nemhauser, George L. Sonstige oth Weismantel, Robert Sonstige oth http://www.sciencedirect.com/science/handbooks/09270507/12 Verlag URL des Erstveröffentlichers Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Discrete optimization Optimisation mathématique Programmation en nombres entiers MATHEMATICS / Optimization bisacsh Integer programming fast Mathematical optimization fast Diskrete Optimierung swd Mathematical optimization Integer programming Diskrete Optimierung (DE-588)4150179-2 gnd |
| subject_GND | (DE-588)4150179-2 |
| title | Discrete optimization |
| title_auth | Discrete optimization |
| title_exact_search | Discrete optimization |
| title_full | Discrete optimization edited by K. Aardal, G.L. Nemhauser and R. Weismantel |
| title_fullStr | Discrete optimization edited by K. Aardal, G.L. Nemhauser and R. Weismantel |
| title_full_unstemmed | Discrete optimization edited by K. Aardal, G.L. Nemhauser and R. Weismantel |
| title_short | Discrete optimization |
| title_sort | discrete optimization |
| topic | Optimisation mathématique Programmation en nombres entiers MATHEMATICS / Optimization bisacsh Integer programming fast Mathematical optimization fast Diskrete Optimierung swd Mathematical optimization Integer programming Diskrete Optimierung (DE-588)4150179-2 gnd |
| topic_facet | Optimisation mathématique Programmation en nombres entiers MATHEMATICS / Optimization Integer programming Mathematical optimization Diskrete Optimierung |
| url | http://www.sciencedirect.com/science/handbooks/09270507/12 |
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