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El. knyga: 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art

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  • Formatas: PDF+DRM
  • Išleidimo metai: 06-Nov-2009
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783540682790
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  • Formatas: PDF+DRM
  • Išleidimo metai: 06-Nov-2009
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783540682790
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50 Years of Integer Programming features talks and panel discussions from the Aussois workshop in 2008, commemorating the cutting-plane algorithm, which transformed the field. Key topics of integer programming are discussed by pioneers of the field.

In 1958, Ralph E. Gomory transformed the field of integer programming when he published a paper that described a cutting-plane algorithm for pure integer programs and announced that the method could be refined to give a finite algorithm for integer programming. In 2008, to commemorate the anniversary of this seminal paper, a special workshop celebrating fifty years of integer programming was held in Aussois, France, as part of the 12th Combinatorial Optimization Workshop.50 Years of Integer Programming offers an account of featured talks and a panel discussion at the Aussois workshop with six pioneers on a DVD Video. It also contains reprints of key historical articles and written versions of survey lectures on six of the hottest topics in the field by distinguished members of the integer programming community. Useful for anyone in mathematics, computer science and operations research, this book exposes mathematical optimization, specifically integer programming and combinatorial optimization, to a broad audience.

Recenzijos

It is a concise, yet voluminous, book giving the theoretical, algorithmic and computational aspects of integer programming. The book provides and serves as an excellent introduction to integer programming. In addition it gives an in depth and great historical perspective of the huge amount of research and development that has taken place in the field of integer programming over a period of 50 years. (Hans W. Ittmann, IFORS News, Vol. 12 (2), June, 2018)









From the reviews:

This volume originates from the 12th Combinatorial Optimization Workshop in Aussois, 2008, where 50 years of integer programming were celebrated. It describes the history and the present state of integer programming. Thevolume consists of four parts . This volume is a precious account of the history and the current state of integer programming. (Rainer Burkard, Mathematical Reviews, Issue 2011 f)

Part I the Early Years
Solution of a Large-Scale Traveling-Salesman Problem
7(22)
George B. Dantzig
Delbert R. Fulkerson
Selmer M. Johnson
The Hungarian Method for the Assignment Problem
29(20)
Harold W. Kuhn
Integral Boundary Points of Convex Polyhedra
49(28)
Alan J. Hoffman
Joseph B. Kruskal
Outline of an Algorithm for Integer Solutions to Linear Programs and An Algorithm for the Mixed Integer Problem
77(28)
Ralph E. Gomory
An Automatic Method for Solving Discrete Programming Problems
105(28)
Ailsa H. Land
Alison G. Doig
Integer Programming: Methods, Uses, Computation
133(66)
Michel Balinski
Matroid Partition
199(20)
Jack Edmonds
Reducibility Among Combinatorial Problems
219(24)
Richard M. Karp
Lagrangian Relaxation for Integer Programming
243(40)
Arthur M. Geoffrion
Disjunctive Programming
283(60)
Egon Balas
Part II From the Beginnings to the State-of-the-Art
Polyhedral Approaches to Mixed Integer Linear Programming
343(44)
Michele Conforti
Gerard Cornuejols
Giacomo Zambelli
Introduction
343(5)
Mixed integer linear programming
343(1)
Historical perspective
344(1)
Cutting plane methods
345(3)
Polyhedra and the fundamental theorem of integer programming
348(11)
Farkas' lemma and linear programming duality
349(3)
Caratheodory's theorem
352(1)
The theorem of Minkowski-Weyl
353(2)
Projections
355(1)
The fundamental theorem for MILP
356(1)
Valid inequalities
357(1)
Facets
357(2)
Union of polyhedra
359(3)
Split disjunctions
362(4)
One-side splits, Chvatal inequalities
365(1)
Gomory's mixed-integer inequalities
366(3)
Equivalence of split closure and Gomory mixed integer closure
368(1)
Polyhedrality of closures
369(4)
The Chvatal closure of a pure integer set
370(1)
The split closure of a mixed integer set
370(3)
Lift-and-project
373(7)
Lift-and-project cuts
374(2)
Strengthened lift-and-project cuts
376(1)
Improving mixed integer Gomory cuts by lift-and-project
377(1)
Sequential convexification
378(2)
Rank
380(7)
Chvatal rank
380(2)
Split rank
382(2)
References
384(3)
Fifty-Plus Years of Combinatorial Integer Programming
387(44)
William Cook
Combinatorial integer programming
387(2)
The TSP in the 1950s
389(8)
Proving theorems with linear-programming duality
397(2)
Cutting-plane computation
399(5)
Jack Edmonds, polynomial-time algorithms, and polyhedral combinatorics
404(6)
Progress in the solution of the TSP
410(5)
Widening the field of application in the 1980s
415(3)
Optimization = Separation
418(2)
State of the art
420(11)
References
425(6)
Reformulation and Decomposition of Integer Programs
431(74)
Francois Vanderbeck
Laurence A. Wolsey
Introduction
431(2)
Polyhedra, reformulation and decomposition
433(8)
Introduction
433(1)
Polyhedra and reformulation
434(6)
Decomposition
440(1)
Price or constraint decomposition
441(23)
Lagrangean relaxation and the Lagrangean dual
443(2)
Dantzig-Wolfe reformulations
445(3)
Solving the Dantzig-Wolfe relaxation by column generation
448(3)
Alternative methods for solving the Lagrangean dual
451(5)
Optimal integer solutions: branch-and-price
456(8)
Practical aspects
464(1)
Resource or variable decomposition
464(7)
Benders' reformulation
465(3)
Benders with integer subproblems
468(2)
Block diagonal structure
470(1)
Computational aspects
471(1)
Extended formulations: problem specific approaches
471(19)
Using compact extended formulations
472(1)
Variable splitting I: multi-commodity extended formulations
473(4)
Variable splitting II
477(3)
Reformulations based on dynamic programming
480(3)
The union of polyhedra
483(2)
From polyhedra and separation to extended formulations
485(2)
Miscellaneous reformulations
487(2)
Existence of polynomial size extended formulations
489(1)
Hybrid algorithms and stronger dual bounds
490(3)
Lagrangean decomposition or price-and-price
490(1)
Cut-and-price
491(2)
Notes
493(12)
Polyhedra
494(1)
Dantzig-Wolfe and price decomposition
494(2)
Resource decomposition
496(1)
Extended formulatations
496(2)
Hybrid algorithms and stronger dual bounds
498(1)
References
498(7)
Part III Current Topics
Integer Programming and Algorithmic Geometry of Numbers
505(56)
Friedrich Eisenbrand
Lattices, integer programming and the geometry of numbers
505(2)
Informal introduction to basis reduction
507(2)
The Hermite normal form
509(6)
Minkowski's theorem
515(3)
The LLL algorithm
518(7)
Kannan's shortest vector algorithm
525(4)
A randomized simply exponential algorithm for shortest vector
529(6)
Integer programming in fixed dimension
535(7)
The integer linear optimization problem
542(3)
Diophantine approximation and strongly polynomial algorithms
545(5)
Parametric integer programming
550(11)
References
556(5)
Nonlinear Integer Programming
561(58)
Raymond Hemmecke
Matthias Koppe
Jon Lee
Robert Weismantel
Overview
562(2)
Convex integer maximization
564(9)
Fixed dimension
564(1)
Boundary cases of complexity
565(3)
Reduction to linear integer programming
568(5)
Convex integer minimization
573(13)
Fixed dimension
573(4)
Boundary cases of complexity
577(3)
Practical algorithms
580(6)
Polynomial optimization
586(18)
Fixed dimension and linear constraints: An FPTAS
587(10)
Semi-algebraic sets and SOS programming
597(4)
Quadratic functions
601(3)
Global optimization
604(7)
Spatial Branch-and-Bound
605(2)
Boundary cases of complexity
607(4)
Conclusions
611(8)
References
612(7)
Mixed Integer Programming Computation
619(28)
Andrea Lodi
Introduction
619(2)
MIP evolution
621(11)
A performance perspective
624(7)
A modeling/application perspective
631(1)
MIP challenges
632(9)
A performance perspective
634(5)
A modeling perspective
639(2)
Conclusions
641(6)
References
642(5)
Symmetry in Integer Linear Programming
647(40)
Francois Margot
Introduction
647(2)
Preliminaries
649(2)
Detecting symmetries
651(2)
Perturbation
653(1)
Fixing variables
653(3)
Symmetric polyhedra and related topics
656(2)
Partitioning problems
658(5)
Dantzig-Wolfe decomposition
659(1)
Partitioning orbitope
660(2)
Asymmetric representatives
662(1)
symmetry breaking inequalities
663(5)
Dynamic symmetry breaking inequalities
664(1)
Static symmetry breaking inequalities
664(4)
Pruning the enumeration tree
668(6)
Pruning with a fixed order on the variables
670(3)
Pruning without a fixed order of the variables
673(1)
Group representation and operations
674(4)
Enumerating all non-isomorphic solutions
678(1)
Furthering the reach of isomorphism pruning
679(1)
Choice of formulation
679(2)
Exploiting additional symmetries
681(6)
References
681(6)
Semidefinite Relaxations for Integer Programming
687(40)
Franz Rendl
Introduction
687(3)
Basics on semidefinite optimization
690(2)
Modeling with semidefinite programs
692(13)
Quadratic 0/1 optimization
692(1)
Max-Cut and graph bisection
693(1)
Stable sets, cliques and the Lovasz theta function
694(2)
Chromatic number
696(2)
General graph partition
698(2)
Generic cutting planes
700(2)
SDP, eigenvalues and the Hoffman-Wielandt inequality
702(3)
The theoretical power of SDP
705(6)
Hyperplane rounding for Max-Cut
705(3)
Coloring
708(3)
Solving SDP in practice
711(10)
Interior point algorithms
711(4)
Partial Lagrangian and the bundle method
715(3)
The spectral bundle method
718(3)
SDP and beyond
721(6)
Copositive and completely positive matrices
721(1)
Copostitive relaxations
722(1)
References
723(4)
The Group-Theoretic Approach in Mixed Integer Programming
727
Jean-Philippe P. Richard
Santanu S. Dey
Introduction
727
The corner relaxation
730
Linear programming relaxations
730
Motivating example
731
Gomory's corner relaxation
737
Group relaxations: optimal solutions and structure
739
Optimizing linear functions over the corner relaxation
739
Using corner relaxations to solve MIPs
744
Extended group relaxations
749
Master group relaxations: definitions and inequalities
754
Groups
754
Master group relaxations of mixed integer programs
756
A hierarchy of inequalities for master group problems
759
Extreme inequalities
766
Extreme inequalities of finite master group problems
766
Extreme inequalities for infinite group problems
769
A compendium of known extreme inequalities for finite and infinite group problems
784
On the strength of group cuts and the group approach
785
Absolute strength of group relaxation
785
Relative strength of different families of group cuts
788
Summary on strength of group cuts
795
Conclusion and perspectives
795
References
797
Part IV DVD-Video/DVD-ROM