Ganzzahlige Programmierung — Ein Überblick

Die nachfolgende Darstellung kann nur einen kurzen und sehr gedrangten Uberblick uber den derzeitigen Stand der Methoden zur ganzzahligen Programmierung geben. Klassische Verfahren (Schnittebenenverfahren, klassische Branch-and-Bound- und Enumerationsverfahren) werden nur kurz und insofern erwahnt, als sie fur die Beschreibung von Modifikationen und neueren Algorithmen wesentlich sind. Methoden und Anwendungen der ganzzahligen Programmierung werden vom Verfasser in einer anderen Arbeit [KORTE, 197l], die demnachst erscheinen wird, ausfuhrlich dargestellt. Dieser Beitrag hat daher nur informierenden Charakter. Im ubrigen sei auf die ausgezeichneten Ubersichtsartikel von BALINSKI [1965] und BALINSKI und SPIELBERG [1969] hingewiesen, die allerdings einige neuere Entwicklungen noch nicht berucksichtigt haben.

[1]  Arthur M. Geoffrion,et al.  An Improved Implicit Enumeration Approach for Integer Programming , 1969, Oper. Res..

[2]  A. M. Geoffrion.  Integer Programming by Implicit Enumeration and Balas’ Method , 1967 .

[3]  Bernhard Korte,et al.  Ein lexikographischer Suchalgorithmus zur Lösung allgemeiner ganzzahliger Programmierungsaufgaben , 1969, Unternehmensforschung.

[4]  Robert Alan Cook.  An algorithm for integer linear programming , 1966 .

[5]  Hans Hermes.  Einführung in die Verbandstheorie , 1955 .

[6]  Frederick S. Hillier,et al.  Efficient Heuristic Procedures for Integer Linear Programming with an Interior , 1969, Oper. Res..

[7]  R. Bellman.  Notes on the theory of dynamic programming IV ‐ Maximization over discrete sets , 1955 .

[8]  R. Gomory.  AN ALGORITHM FOR THE MIXED INTEGER PROBLEM , 1960 .

[9]  Michel Balinski,et al.  Integer Programming: Methods, Uses, Computations , 1965 .

[10]  L. G. Mitten.  Branch-and-Bound Methods: General Formulation and Properties , 1970, Oper. Res..

[11]  S. Vajda,et al.  Integer Programming and Network Flows , 1970 .

[12]  K. Spielberg.  Plant Location with Generalized Search Origin , 1969 .

[13]  W. C. Healy.  Multiple Choice Programming A Procedure for Linear Programming with Zero-One Variables , 1964 .

[14]  R. D. Young.  A Simplified Primal (All-Integer) Integer Programming Algorithm , 1968, Oper. Res..

[15]  G. H. Bradley.  Equivalent Integer Programs and Canonical Problems , 1971 .

[16]  Egon Balas.  Intersection Cuts from Maximal Convex Extensions of the Ball and the Octahedron. , 1970 .

[17]  Frederick S. Hillier.  A Bound-and-Scan Algorithm for Pure Integer Linear Programming with General Variables , 1969, Oper. Res..

[18]  R. Gomory.  Some polyhedra related to combinatorial problems , 1969 .

[19]  E. Balas.  An Additive Algorithm for Solving Linear Programs with Zero-One Variables , 1965 .

[20]  E. M. L. Beale,et al.  Survey of Integer Programming , 1965 .

[21]  G. Dantzig.  Note on solving linear programs in integers , 1959 .

[22]  Egon Balas,et al.  Discrete Programming by the Filter Method , 1967, Oper. Res..

[23]  J. F. Benders.  Partitioning procedures for solving mixed-variables programming problems , 1962 .

[24]  Raoul J. Freeman.  Letter to the Editor - Computational Experience with a "Balasian" Integer Programming Algorithm , 1966, Oper. Res..

[25]  A. Land,et al.  An Automatic Method for Solving Discrete Programming Problems , 1960, 50 Years of Integer Programming.

[26]  Robert B. Wilson.  Letter to the Editor - Stronger Cuts in Gomory's All-Integer Integer Programming Algorithm , 1967, Oper. Res..

[27]  Fred W. Glover.  Cut search methods in integer programming , 1972, Math. Program..

[28]  R. P. Dilworth.  Review: G. Birkhoff, Lattice theory , 1950 .

[29]  Clifford C. Petersen,et al.  Computational Experience with Variants of the Balas Algorithm Applied to the Selection of R&D Projects , 1967 .

[30]  Richard D. Young.  A primal (all-integer) integer programming algorithm , 1965 .

[31]  Harvey M. Salkin.  Letter to the Editor - On the Merit of the Generalized Origin and Restarts in Implicit Enumeration , 1970, Oper. Res..

[32]  Jeremy F. Shapiro,et al.  Group Theoretic Algorithms for the Integer Programming Problem II: Extension to a General Algorithm , 1968, Oper. Res..

[33]  Solomon W. Golomb,et al.  Backtrack Programming , 1965, JACM.

[34]  Samsão Woiler,et al.  Implicit enumeration algorithms for discrete optimization problems , 1968 .

[35]  F. Glover.  A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problem , 1965 .

[36]  Fred W. Glover,et al.  A New Foundation for a Simplified Primal Integer Programming Algorithm , 1968, Oper. Res..

[37]  R. J. Dakin,et al.  A tree-search algorithm for mixed integer programming problems , 1965, Comput. J..

[38]  Norman J. Driebeek.  An Algorithm for the Solution of Mixed Integer Programming Problems , 1966 .

[39]  Lawrence J. Watters.  Letter to the Editor - Reduction of Integer Polynomial Programming Problems to Zero-One Linear Programming Problems , 1967, Oper. Res..

[40]  Egon Balas,et al.  THE INTERSECTION CUT - A NEW CUTTING PLANE FOR INTEGER PROGRAMMING. , 1969 .

[41]  Ralph E. Gomory,et al.  An algorithm for integer solutions to linear programs , 1958 .

[42]  R. D. Young.  Hypercylindrically Deduced Cuts in Zero-One Integer Programs , 1971, Oper. Res..

[43]  Egon Balas,et al.  Letter to the Editor - A Note on the Branch-and-Bound Principle , 1968, Oper. Res..

[44]  E. L. Lawler,et al.  A Method for Solving Discrete Optimization Problems , 1966, Oper. Res..

[45]  Fred W. Glover,et al.  Letter to the Editor - Stronger Cuts in Integer Programming , 1967, Oper. Res..

[46]  Fred Glover,et al.  A Note on the Additive Algorithm of Balas , 1965 .

[47]  J. Shapiro.  Dynamic Programming Algorithms for the Integer Programming Problem—I: The Integer Programming Problem Viewed as a Knapsack Type Problem , 1968 .

[48]  F. Glover.  Generalized Cuts in Diophantine Programming , 1966 .

[49]  Andrew B. Whinston,et al.  A New Approach to Discrete Mathematical Programming , 1968 .

[50]  Egon Balas,et al.  An Intersection Cut from the Dual of the Unit Hypercube , 1971, Oper. Res..

[51]  D. J. Laughhunn.  Quadratic Binary Programming with Application to Capital-Budgeting Problems , 1970, Oper. Res..

[52]  A. J. Hoffman,et al.  On the convergence of an integer‐programming process , 1963 .

[53]  Bernhard Fleischmann.  Letter to the Editor - Computational Experience with the Algorithm of Balas , 1967, Oper. Res..

[54]  Andrew Whinston,et al.  An Algorithm for the Quadratic Assignment Problem , 1970 .

[55]  E. L. Lawler,et al.  Branch-and-Bound Methods: A Survey , 1966, Oper. Res..

[56]  George L. Nemhauser,et al.  Deep cuts in integer programming , 1971 .