Skip to main content
SpringerLink
Log in

Linear Programming

  • Chapter
  • First Online:
Introduction to Combinatorial Optimization

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 196))

  • 1061 Accesses

Abstract

Linear programming (LP) is an important combinatorial optimization problem, and in addition, it is an important tool to design and to understand algorithms for other problems. In this chapter, we introduce LP theory starting from Simplex Algorithm, which is an incremental method.

Our intuition about the future is linear.

—Ray Kurzweil

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. D. Bayer and J.C. Lagarias: The non-linear geometry of linear programming, I. Affine and projective scaling trajectories, II. Legendre transform coordinates, III. Central trajectories, Preprints, AT&T Bell Laboratories (Murray Hill, NJ, 1986).

    Google Scholar 

  2. E.M. Beale: Cycling in dual simplex algorithm, Navel Research Logistics Quarterly 2: 269–276 (1955).

    Article  MathSciNet  Google Scholar 

  3. Robert G. Bland: New finite pivoting rules for the simplex method, Mathematics of Operations Research 2 (2): 103–107 (1977).

    Article  MathSciNet  Google Scholar 

  4. A. Charnes: Optimality and degeneracy in linear programming, Econometrica, 2: 160–170 (1952).

    Article  MathSciNet  Google Scholar 

  5. G.B. Dantzig: Application of the simplex method to a transportation problem, in: Activity Analysis of Production and Allocation, (Cowles Commission Monograph 13), T.C. Koopmans (ed.), John-Wiley, New York, pp. 359–373, 1951.

    Google Scholar 

  6. G.B. Dantzig: Maximization of a linear function of variables subject to linear inequalities, Chap. XXI of Activity Analysis of Production and Allocation, (Cowles Commission Monograph 13), T.C. Koopmans (ed.), John-Wiley, New York, 1951, pp. 339–347.

    Google Scholar 

  7. G.B. Dantzig: A. Orden, P. Wolfe: Note on linear programming, Pacific J. Math, 5: 183–195 (1955).

    Google Scholar 

  8. J. Edmonds: Maximum matching and a polyhedron with 0, 1-vertices, Journal of Research National Bureau of Section B, 69: 125–130 (1965).

    MathSciNet  MATH  Google Scholar 

  9. J. Edmonds: Minimum partition of a matroid into independent subsets, Journal of Research National Bureau of Section B, 69: 67–72 (1965).

    MathSciNet  MATH  Google Scholar 

  10. J. Edmonds: Paths, trees and flowers, Canadian Journal of Mathematics, 17: 449–467 (1965).

    Article  MathSciNet  Google Scholar 

  11. J. Edmonds: Optimum branchings, Journal of Research National Bureau of Section B, 71: 233–240 (1967).

    MathSciNet  MATH  Google Scholar 

  12. J. Edmonds: Submodular functions, matroids, and certain polyhedrons, in: Combinatorial Structure and Their Applications (R. Guy, H. Hanani, N. Sauer, J. Schönheim, eds.) Gordon and Breach, New York, pp. 69–87, 1970.

    Google Scholar 

  13. J. Edmonds: Edge-disjoint branchings, in Combinatorial Algorithms (R. Rustin, ed.) Algorithmics Press, New York, pp. 91–96, 1973.

    Google Scholar 

  14. L.R. Ford, D.R. Fulkerson: Maximal flow through a network, Canadian Journal of Mathematics, 8: 399–404 (1956).

    Article  MathSciNet  Google Scholar 

  15. L.R. Ford, D.R. Fulkerson: Solving the transportation problem, Manamental Science, 3: 24–32 (1956-57).

    Google Scholar 

  16. L.R. Ford, D.R. Fulkerson: A simple algorithm for finding maximal network flow and an application to Hitchcock problem, Canadian Journal of Mathematics, 9: 210–218 (1957).

    Article  MathSciNet  Google Scholar 

  17. C. C. Gonzaga: Polynomial affine algorithms for linear programming, Mathematical Programming, 49: 7–21 (1990).

    Article  MathSciNet  Google Scholar 

  18. C. Gonzaga: An algorithm for solving linear programming problems in O(n3L) operations, in: N. Megiddo, ed., Progress in Mathematical Programming: Interior-Point and Related Methods, pp. 1–28, Springer, New York, 1988.

    Google Scholar 

  19. C. Gonzaga: Conical projection algorithms for linear programming, Mathematical Programming, 43: 151–173 (1989).

    Article  MathSciNet  Google Scholar 

  20. A.J. Hoffman: Some recent applications of the theory of linear inequalities to extremal combinatorial analysis, in Combinatorial Analysis (Yew York, 1958; R. Bellman, M. Hall, Jr, eds.), American Mathematical Society, Providence, Rhode Islands, pp. 113–127, 1960.

    Google Scholar 

  21. L.V. Kantorovich: A new method of solving some classes of extremal problems, Doklady Akad Sci SSSR, 28: 211–214 (1940).

    MathSciNet  Google Scholar 

  22. N. Karmakkar: A new polynomial-time algorithm for linear programming, Proceedings, 16th Annual ACM Symposium on the Theory of Computing, pp. 302–311, 1984.

    Google Scholar 

  23. L.G. Khachiyan: A polynomial algorithm for linear programming, Doklad. Akad. Nauk. USSR Sec., 244: 1093–1096 (1979).

    MathSciNet  MATH  Google Scholar 

  24. M. Kojima, S. Mizuno and A. Yoshise: A primal-dual interior point method for linear programming, in: Progress in Mathematical Programming: Interior-Point and Related Methods (N. Megiddo, ed.), pp. 29–48, (Springer, New York, 1988).

    Google Scholar 

  25. H.W. Kuhn: The Hungarian method for the assignment problem, Naval Research Logistics Quarterly, 2: 83–97 (1955).

    Article  MathSciNet  Google Scholar 

  26. H.W. Kuhn: Variants of the Hungarian method for assignment problems, Naval Research Logistics Quarterly, 3: 253–258 (1956).

    Article  MathSciNet  Google Scholar 

  27. N. Megiddo and M. Shub: Boundary behaviour of interior point algorithms in linear programming, Research Report RJ 5319, IBM Thomas J. Watson Research Center (Yorktown Heights, NY, 1986).

    Google Scholar 

  28. R.C. Monteiro and I. Adler: An O(n 3 L) primal-dual interior point algorithm for linear programming, Manuscript, Department of Industrial Engineering and Operations Research, University of California (Berkeley, CA, 1987).

    Google Scholar 

  29. J. Renegar: A polynomial-time algorithm based on Newton’s method for linear programming, Mathematical Programming, 40: 59–94 (1988).

    Article  MathSciNet  Google Scholar 

  30. A. Schrijver: Theory of Linear and Integer Programming, (Wiley, Chichester, 1986).

    MATH  Google Scholar 

  31. Alexander Schrijver: Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics. 24. (Springer, 2003).

    Google Scholar 

  32. M. Todd and B. Burrell: An extension of Karmarkar’s algorithm for linear programming using dual variables, Algorithmica, 1: 409–424 (1986).

    Article  MathSciNet  Google Scholar 

  33. M.J. Todd and Y. Ye: A centered projective algorithm for linear programming, Technical Report 763, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY, 1987).

    Google Scholar 

  34. Pravin M. Vaidya: An algorithm for linear programming which requires O(((m + n)n 2 + (m + n)1.5 n)L) arithmetic operations, Mathematical Programming, 47: 175–201 (1990).

    Article  MathSciNet  Google Scholar 

  35. Jianzhong Zhang, Shaoji Xu: Linear Programming, (Schiece Press, 1987).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Texas, Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. Department of Industrial & Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos Pardalos

  3. University of Chinese Academy of Sciences Academy of Math and System Science, Chinese Academy of Sciences, Beijing, China

    Xiaodong Hu

  4. Department of Computer Science, University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

Authors
  1. Ding-Zhu Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Panos Pardalos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaodong Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Weili Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Du, DZ., Pardalos, P., Hu, X., Wu, W. (2022). Linear Programming. In: Introduction to Combinatorial Optimization. Springer Optimization and Its Applications, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-031-10596-8_6

Download citation

Publish with us

Policies and ethics

Search

Navigation