Linear programming: A survey of general purpose algorithms

AIIE Transactions

Volumn 7, Issue 3, 1975, Pages 212-221

  Crisp, Robert M ^b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSITY OF ARKANSAS   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL PROGRAMMING, LINEAR;

EID: 0016556661     PISSN: 05695554     EISSN: None     Source Type: Journal    
DOI: 10.1080/05695557508975005     Document Type: Article

References (88)

1. Dantzig, George B., private correspondence, Stanford University, Stanford, California (August 8, 1974).
2. Geoffrion, A. M., “A Guided Tour of Recent Practical Advances in Integer Linear Programming,” Working Paper No. 220, Western Management Science Institute; UCLA, Los Angeles, California (September 1974).
3. Zionts, Stanley, private correspondence, European Institute for Advanced Studies in Management, Brussels (October 29, 1974).
4. Chames, A. and W. W. Cooper, Management Models and Industrial Applications of Linear Programming, V. I and II, 3 ohn Wiley and Sons (1961).
5. Dantzig, G. B., Linear Programming and Extensions, Princeton University Press (1963).
6. Gass, S. I., Linear Programming, Methods and Applications, McGraw-Hill Book Company, Inc. (1958).
7. Hadley, G., Linear Programming, Addison-Wesley Publishing Company, Inc. (1952).
8. Llewellyn, R. W., Linear Programming, Holt, Rinehart and Winston, Inc. (1966).
9. Simmonard, M., Linear Programming, Prentice-Hall, Inc. (1966).
10. Spivey, W. A. and R. M. Thrall, Linear Optimization, Holt, Rinehart and Winston, Inc. (1970).
11. Zionts, S., Linear and Integer Programming, Prentice-Hall, Inc. (1974).
12. Bennett, J. M., “An Approach to Some Structural Linear Programming Problems,” Operations Research, 14, pp. 636645 (1966).
13. Charnes, A. and C. E. Lemke, “Computational Theory of Linear Programming I: The Bounded Variables Problem,” ONR Research Memo, No. 10, Graduate School of Industrial Administration, Carnegie-Mellon University (1954).
14. Cord, T., “Generalized Upper Bounds with Upper Bounded Variables,” IBM New York Scientific Center Report No. 320-2962 (1969).
15. Dantzig, G. B., “Upper Bounds, Secondary Constraints, and Block Triangularity,” Econometrica, 23 2, pp. 174-183 (1955).
16. Dantzig, G. B. and P. Wolfe, “Decomposition Principle for Linear Programs,” Operations Research, 8, pp. 101-111 (1960).
17. Dantzig, G. B. and R. M. Van Slyke, “Generalized Upper Bounding Techniques,” Journal of Computer and System Sciences, 1, pp. 213-226 (1967).
18. Glover, Fred and D. Korney, D. Klingman, A. Naplier, “A Computation Study on Start Procedures, Basis Change Criteria, and Solution Algorithms for Transportation Problems,” Afenagement Science, 20 5, 793-813 (January 1974).
19. Grigoriadis, M. D., “A Dual Generalized Upper Bounding Technique,” Management Science, 17 5, pp. 269-285 (January 1971).
20. Lemke, C. E., “The Dual Method for Solving the Linear Programming Problem,” Naval Research Logistics Quarterly, 1, pp. 36-47 (1954).
21. Wagnei, H. M., “The Dual Simplex Algorithm for Bounded Variables,” Naval Logistics Research Quarterly, 5 3, pp. 257261 (1958).
22. Balinski, M. L. and R. E. Gomory, “A Mutual Primal Dual Simplex Method” in Recent Advances in Mathematical Projamming, Ed. R. L. Graves and P. Wolfe, McGraw-Hill (1963).
23. Dantzig, G. B., L. R. Ford, and D. R. Fulkerson, “A Primal- Dual Algorithm for Linear Programs,” in Linear Equalities tnd Related Systems, Kuhn and Tucker, Eds., Princeton University Press (1956).
24. Garfinkel, Robert S. and P. L. Yu, “A Primal-Dual Surrogate Simplex Algorithm: Comparison with the Criss-Cross Method,” CSS 73-12 (May 1974).
25. Harris, M. Y., “A Mutual Primal-Dual Linear Programming Algorithm,” Naval Research Logistics Quarterly, 19 2, pp. 199-206 (June 1970).
26. Kuhn, H. W., “The Hungarian Method for the Assignment Problem,” Naval Research Logistics Quarterly 2, pp. 83-97, (1955).
27. Mills, R., An Evaluation of the Primal-Dual Algorithm for Linear Programming, Masters Thesis, MIT, Cambridge, Massachusetts (1960).
28. Mueller, R. K. and L. Cooper, “A Comparison of the Primal Simplex and Primal-Dual Algorithms for Linear Programming,” Communications of the ACM, 8 11, pp. 682-686 (1965).
29. Talacko, J. and R. T. Rockefeller, “Linear Programming Solved by Symmetric Methods,” Marquette Business Review, 4 3, pp. 32-48 (1960).
30. Talacko, J. V. and R. T. Rockefeller, “A Compact Simplex Algorithm and a Symmetric Algorithm for General Linear Programs,” Unpublished paper, Marquette University, August (1960).
31. Talacko, J., “Natural Computational Method of Two-Person Zero Sum Games,” Travajos de Estadística y de Investigación Operativa, 14 1, 2, pp. 17-60 (1963).
32. Wolfe, P., “The Composite Simplex Algorithm,” SIAM Review, 7 1, pp. 42-54 (1965).
33. Zionts, S., “The Criss-Cross Method for Solving Linear Programming Problems,” Management Science, 15 7, pp. 426445 (1969).
34. Zionts, S., “Some Empirical Tests of the Criss-Cross Method,” Management Science, 19 4, pp. 406-410 (1972).
35. Ben-Israel, A., et al., “On the Explicit Solution of a Special Class of Linear Economic Models,” Operations Research, 18, pp. 462-470 (1970).
36. Ben-Israel, A. and A. Charnes, “An Explicit Solution of a Special Class of Linear Programming Problems,” Operations Research, 16, pp. 1166-1175 (1968).
37. Ben-Israel, A. and P. D. Robers, “A Decomposition Method for Interval Linear Programming,”Management Science, 165, pp. 374-387 (1968).
38. Boullion, Thomas L. and Patrick L. Odell, An Introduction to the Theory of Generalized Matrix Invertability, Texas Center for Research, Austin, Texas (1966).
39. Charnes, A. and A. Ben-Israel, “Contributions to the Theory of Generalized Inverses,” J. Soc. Indust. Appl. Math, 11, 3, pp. 667-699 (1963).
40. Cline, R. E., “Representations for the Generalized Inverse of a Partitioned Matrix,” Journal Soc. Indust. Appl. Math., 12, pp. 588^600 (1964).
41. Cline, R. E., Representations for the Generalized Inverse of Matrices with Applications in Linear Programming. Doctoral Thesis, Purdue University, Lafayette, Indiana (1963).
42. Gay, D. M., “On Scolnik’s Proposed Polynomial Time Linear Programming Algorithm,” TR73-190, Computer ScienceDepartment, Cornell University (June 1974).
43. Gay, D. M., “More Remarks on Scolnik’s Approach to Linear Programming,” TR74-207, Computer Science Department, Cornell University (1974).
44. Gay, D. M., Personal conversations (October 1974).
45. Greville, T. N. E. and A. Ben-Israel, Generalized Inverses: Theory and Applications, Wiley-Interscience (1974).
46. Pyle, L. D., The Generalized Inverse in Linear Programming, Doctoral Thesis, Purdue University, Lafayette, Indiana (1960).
47. Pyle, L. D., “Generalized Inverse Computations Using the Gradient Projection Method,” Journal of the ACM, 114, pp. 422-28 (1964).
48. Pyle, L. D., “A Generalized Inverse e-Algorithm for Intersection Projection Matrices, with Applications,” Numerische Mathematik, 10, pp. 86-102 (1967).
49. Pyle, L. D., The Generalized Inverse in Linear Programming- Basic Structure,” SIAM Journal of Applied Mathematics, 22, pp. 335-355 (1972).
50. Pyle, L. D. and R. E. Cline, “The Generalized Inverse in Linear Programming-An Intersection Projection Method and the Solution of a Class of Structured Linear Programming Problems,” SIAM Journal of Applied Mathematics, 24 3, pp. 338-351 (1973).
51. Pyle, L. D. and R. E. Cline, “The Generalized Inverse in Linear Programming—Interior Gradient Projection Methods,” SIAM Journal of Applied Mathematics, 24 4, pp. 511-534 (1973).
52. Rao, C. R. and Sujit K. Mitro, Generalized Inverse of Matrices andlts Applications, Wiley (1971).
53. Robers, P. D. and A. Ben-Israel, “A Suboptimization Method for Interval Linear Programming,” Linear Algebra and Its Applications, 3 3, pp. 383-405 (1970).
54. Robers, P. D., “Some Numerical Experiments Comparing the Numerical Efficiency of SIMPLEX and SUBOPT,” Staff Memorandum, Research Analysis Corporation, McClean, Virginia.
55. Rockett, Richard L., “Optimal Solutions to Classes of Linear Programming Problems Using Modified Least Squares Techniques,” Unpublished Masters Thesis, The University of Oklahoma, Norman, Oklahoma (1974).
56. Rosen, J. B., “The Gradient Projection Method for NonLinear Programming. Part 1: Linear Constraints,” J. Soc. Indust. Appl. Math., 8, pp. 181-217 (1960).
57. Scolnik, H. D., “A New Approach to Linear Programming,” SIGMAP Newsletter (November 1973).
58. Scolnik, H. D., “Letter to SIGMAP Members,” SIGMAP Newsletter, p. 124 (April 1974).
59. Scolnik, H. D., Untitled Preliminary Report released at Eighth International Symposium on Mathematical Programming.
60. Sposito, V. A., “Solutions of a Special Class of Linear Programming Problems,” Operations Research, 21, pp. 386-388 (1973).
61. White, W. W., “A Commentary on the ‘New Approach to L.P.’” SIGMAP Newsletter, No. 16 (April 1974).
62. Wylie, C. R., It., Advanced Engineering Mathematics, McGraw- Hill, pp. 458-459(1966).
63. Zlobec, A. and A. Ben-Israel, “On Explicit Solutions of Interval Programs,” Israel J. Mathematics, 8, pp. 12-22 (1970).
64. Cottle, R. W. and G. B. Dantzig, “Complementary Pivot Theory of Mathematical Programming,” Linear Algebra and Its Applications, 1, pp. 103-125 (1968).
65. Cottle, R. W. and G. B. Dantzig, “Positive (Semi-)Definite Programming,” in Nonlinear Programming, J. Abadie, Ed., North-Holland, Amsterdam, pp. 55-73 (1967).
66. Dittmann, J. P., Surrogated Linear and Quadratic Programming, Ph.D. Dissertation, University of Missouri-Columbia (1973).
67. Dittmann, J. P., and Glenn E. Staats, “Surrogated Linear Programming,” AIIE Transactions, 5 4, pp. 327-333 (December 1973).
68. Eaves, B. Curtis, “The Linear Complementarity Problem in Mathematical Programming,” Management Science, 17, pp. 612-624 (1971).
69. Glover, F., “A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problems,” Operations Research, 13, pp. 879-919 (1965).
70. Graves, R. L., “A Principal Pivoting Simplex Algorithm for Linear and Quadratic Programming,” Operations Research, 15, pp. 482-495 (1967).
71. Greenberg, H. J. and W. P. Pierskalla, “Surrogate Mathematical Programming,” Operations Research, 18, pp. 924-939 (1970).
72. Lemke, C. E., “Bimatrix Equilibrium Points and Mathematical Programming,” Management Science, 11, pp. 681-689 (1965).
73. Lemke, C. E. and J. T. Howson, Jr., “Equilibrium Points of Bimatrix Games,” Journal of the Society of Industrial and Applied Mathematics, 12 2, pp. 413-423 (1964).
74. Levinson, N. L., “Linear Programming in Complex Space,” Journal of Mathematical Analysis and Applications, 14, pp. 44-62 (1966).
75. McCallum, C. J., Jr., “Existence Theory for the Complex Linear Complementarity Problem,” Journal of Mathematical Analysis and Applications, 40 3, pp. 738-762 (1972).
76. McCallum, C. J., “Solution of the Complex Linear Complementarity Problem,” Journal of Mathematical Analysis and Applications, 44 3, pp. 643-660 (1973).
77. Ravindran, A., “A Comparison of the Primal-Simplex and Complementary Pivot Method for Linear Programming,” Opsearch, 7, pp. 241-262 (1960).
78. Saigal, R., “On the Class of Complementary Cones and Lemke’s Algorithm,” SIAM J. Appl. Math., 23 1, pp. 46-60.
79. Saksena, C. P. and A. J. Cole, “The Bounding Hyperplane Method of Linear Programming,” Operations Research, 19 1, pp. 1-18 (1971).
80. Staats, Glenn E., Computational Aspects of Geometric Programming with Degrees of Difficulty, Ph.D. Dissertation, University of Texas at Austin (1970).
81. Tucker, A. W., “Game Theory and Programming,” National Science Foundation Summer Mathematics Institute Notes (Stillwater: Department of Mathematics, The Oklahoma Agricultural and Mechanical College, 1955).
82. Arnofsky, J. S. (ed.), Progress in Operations Research, Volume III, Wiley, New York (1969).
83. Lasdon, L. S., Optimization Theory for Large Systems, The Macmillan Company, New York (1970).
84. Orchard-Hays, W., Advanced Linear Programming Computing Techniques, McGraw-Hill, New York, 1968.
85. Tomlin, J. A., “Maintaining a Sparse Inverse in the Simplex Method,” IBM Journal of Research and Development, 16 4, pp. 415-423 (1972).
86. White, W. W., “On the Computational Status of Mathematical Programming,” IBM-New York Scientific Center, Data Processing Center, Technical Report No. 320-2990 (June 1970).
87. ICP Quarterly, pp. 305-310 (January 1973).
88. Mathematical Programming System-Extended (MPSX) Manual, SH20-0968-1, International Business Machines (IBM).