# Computational Methods For Linear Programming

• D. F. Shanno
Part of the NATO ASI Series book series (ASIC, volume 434)

## Abstract

The paper examines two methods for the solution of linear programming problems, the simplex method and interior point methods derived from logarithmic barrier methods. Recent improvements to the simplex algorithm, including primal and dual steepest edge algorithms, better degeneracy resolution, better initial bases and improved linear algebra are documented. Logarithmic barrier methods are used to develop primal, dual, and primal-dual interior point methods for linear programming. The primal-dual predictor-corrector algorithm is fully developed. Basis recovery from an optimal interior point is discussed, and computational results are given to document both vast recent improvement in the simplex method and the necessity for both interior point and simplex methods to solve a significant spectrum of large problems

## Keywords

Linear Programming Problem Simplex Method Interior Point Method Simplex Algorithm Barrier Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

1. [1]
Adler, I., Karmarkar, N. K., Resende, M. G. C., and Veiga, G. (1989) An implementation of Karmarkar’s algorithm for linear programming, Mathematical Programming 44, 297–335
2. [2]
Barnes, E. R. (1986) A variation on Karmarkar’s algorithm for solving linear programming problems, Mathematical Programming 36, 174–182
3. [3]
Bixby, R. E. (1992) Implementing the simplex method: The initial basis,ORSA Journal on Computing 4, 267–284
4. [4]
Bixby, R. E., Gregory, J. W., Lustig, I. J., Marsten, R. E., and Shanno, D. F. (1992) Very large-scale linear programming: A case study in combining interior point and simplex methods,Operations Research 40, 885–897
5. [5]
Bixby, R. E. and Lustig, I. J. (1993) An implementation of a strongly polynomial time algorithm for basis recovery, In preparation Department of Computational and Applied Mathematics, Rice University Houston, TX, USAGoogle Scholar
6. [6]
CPLEX Optimization, Inc. (1993) Using the CPLEX™ Callable Library and CPLEX™ Mixed Integer Library Incline Village, NevadaGoogle Scholar
7. [7]
Dikin, I.I. (1967) Iterative solution of problems of linear and quadratic programming Doklady Akademii Nauk SSSR 174 747–748 Translated in Soviet Mathematics Doklady 8 674–675, 1967
8. [8]
Dikin, I. I. (1974) On the convergence of an iterative process,Upravlyaemye Sistemi 12, 54–60 (In Russian).Google Scholar
9. [9]
Duff, I. S., Erisman, A., and Reid, J. (1986) Direct Methods for Sparse Matrices, Clarendon Press, Oxford, England
10. [10]
Fiacco, A. V. and McCormick, G. P. (1968) Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley & Sons, New York
11. [11]
Forrest, J. J. H. and Tomlin, J. A. (1972) Updating triangular factors of the basis matrix to maintain sparsity in the product form simplex method,Mathematical Programming 2, 263–278
12. [12]
Forrest, J. J. H. and Tomlin, J. A. (1990) Vector processing in simplex and interior methods for linear programming, Annals of Operations Research 22, 71–100
13. [13]
Forrest, J. J. H. and Goldfarb, D. (1992) Steepest edge simplex algorithms for linear programming,Mathematical Programming 57, 341–374
14. [14]
Frisch, K. R. (1955) The logarithmic potential method for convex programming, Unpublished manuscript, Institute of Economics, University of Oslo Oslo, NorwayGoogle Scholar
15. [15]
George, A. and Liu, J. (1981) Computer Solution of Large Sparse Positive Definite Systems, Prentice Hall, Englewood Cliffs, NJ
16. [16]
Gill, P. E., Murray, W., Saunders, M. A., Tomlin, J. A., and Wright, M. H. (1986) On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method,Mathematical Programming 36, 183–209
17. [17]
Gill, P. E., Murray, W., Saunders, M. A., and Wright, M. H. (1989) A practical anticycling procedure for linearly constrained optimization,Mathematical Programming 45, 437–474
18. [18]
Goldfarb, D. and Reid, J. K. (1977) A practical steepest-edge simplex algorithm,Mathematical Programming 12, 361–371
19. [19]
Gonzaga, C. C. (1992) Path following methods for linear programming,SIAM Review 34(2), 167–227
20. [20]
Harris, P. M. J. (1973) Pivot selection methods of the Devex lp code,Mathematical Programming 5, 1–28
21. [21]
International Business Machines Corporation (1991) Optimization Subroutine Library Guide and Reference, Release 2Google Scholar
22. [22]
Karmarkar, N. K. (1984) A new polynomial-time algorithm for linear programming,Combinatorica 4, 373–395
23. [23]
Kojima, M., Mizuno, S., and Yoshise, A. (1989) A primal-dual interior point algorithm for linear programming, In N. Megiddo, (ed.), Progress in Mathematical Programming: Interior Point and Related Methods, pp. 29–47 Springer Verlag New YorkGoogle Scholar
24. [24]
Liu, J. (1985) Modification of the minimum-degree algorithm by multiple elimination,ACM Transactions on Mathematical Software 11, 141–153
25. [25]
Lustig, I. J., Marsten, R. E., and Shanno, D. F. (1992) On implementing Mehrotra’s predictor-corrector interior point method for linear programming,SIAM Journal on Optimization 2, 435–449
26. [26]
Lustig, I. J., Marsten, R. E., and Shanno, D. F. (1992) The interaction of algorithms and architectures for interior point methods, In P. M. Pardalos, (ed.), Advances in Optimization and Parallel Computing, pp. 190–205 North-Holland Amsterdam, The NetherlandsGoogle Scholar
27. [27]
Lustig, I. J., Marsten, R. E., and Shanno, D. F. (1992) Interior point methods: Computational state of the art, Technical Report School of Engineering and Applied Science, Dept. of Civil Engineering and Operations Research, Princeton University Princeton, NJ 08544, USA Also available as RUTCOR Research Report RRR 41–92, RUTCOR, Rutgers University, New Brunswick, NJ, USA. To appear in ORSA Journal on Computing Google Scholar
28. [28]
McShane, K. A., Monma, C. L., and Shanno, D. F. (1989) An implementation of a primal-dual interior point method for linear programming,ORSA Journal on Computing 1, 70–83
29. [29]
Megiddo, N. (1989) Pathways to the optimal set in linear programming, In N. Megiddo, (ed.), Progress in Mathematical Programming: Interior Point and Related Methods, pp. 131–158 Springer Verlag New YorkGoogle Scholar
30. [30]
Megiddo, N. (1991) On finding primal-and dual-optimal bases,ORSA Journal on Computing 3, 63–65
31. [31]
Mehrotra, S. (1992) On the implementation of a primal-dual interior point method,SI AM Journal on Optimization 2(4), 575–601
32. [32]
Orchard-Hays, W. (1968) Advanced Linear Programming Computing Techniques, McGraw-Hill, New York, NY, USAGoogle Scholar
33. [33]
Suhl, U. H. and Suhl, L. M. (1990) Computing sparse LU factorizations for large-scale linear programming bases,ORSA Journal on Computing 2, 325–335
34. [34]
Vanderbei, R. J., Meketon, M. S., and Freedman, B. A. (1986) A modification of Karmarkar’s linear programming algorithm,Algorithmica 1(4), 395–407

## Copyright information

© Kluwer Academic Publishers 1994

## Authors and Affiliations

• D. F. Shanno
• 1
1. 1.Rutgers Center for Operations ResearchRutgers UniversityUSA