Advertisement

Interior point approach to linear programming: theory, algorithms & parametric analysis

Part of the Mathematics and Its Applications book series (MAIA, volume 81)

Abstract

The classical theory of linear programming strongly depends on the fact that among the optimal solutions of an LP-problem there is always a vertex solution. In many situations the analysis is complicated by the fact that this vertex solution may not be unique. The recent research in the field of interior point methods for LP has made clear that every (solvable) LP-problem has a unique socalled central solution, namely the analytic center of the optimal facet of the problem. In this paper we reconsider the theory of LP by using central solutions. The analysis is facilitated by the unicity of the central solution of an LP-problem. Starting from scratch, using an elementary result from calculus, we present new proofs of the fundamental results of LP. These include the existence of a strictly complementary solution, and the strong duality theorem for LP. The proofs are simpler and often more natural than the ones currently known. It turns out that the central solution of an LP-problem is the limit point of the socalled central path of the problem. Based on this observation an algorithm will be derived which approximately follows the central path. The output of this algorithm is an approximation of the central solution. It also gives us the optimal partition of the problem. We finally deal with the topic of parametric analysis. So we investigate the dependence of the central solution on the right hand side coefficients and/or the coefficients in the objective vector. It turns out that also from the parametric point of view the interior point approach is more natural than the usual simplex based approach.

Keywords

Interior Point Feasible Region Interior Point Method Central Solution Central Path 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Anstreicher, K.M (1985), A Monotonic Projective Algorithm for Fractional Linear Programming Algorithmica 1 483–498.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Anstreicher, K.M., Bosch, R.A. (1988), Long Steps in a O ( n 3 L )Algorithm for Linear Programming, Preprint, Yale School of Organization and Management, New Haven, CT.Google Scholar
  3. [3]
    Barnes, E.R. (1986), A Variation on Karmarkar’s Algorithm for Solving Linear Programming Problems Mathematical Programming 36, 174–182.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Dikin, I.I. (1967), Iterative Solution of Problems of Linear and Quadratic Programming Doklady Akademiia Nauk SSSR 174, 747–748.MathSciNetGoogle Scholar
  5. [5]
    De Ghellinck, G. and Vial, J.-Ph. (1986), A Polynomial Newton Method for Linear Programming Algorithmica 1, 425–453.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Freund, R.M. (1988), Polynomial-Time Algorithms for Linear Programming Based only on Primal Scaling and Projected Gradients of a Potential Function, Working Paper 182–88, Massachusetts Institute of Technology, Massachusetts.Google Scholar
  7. [7]
    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.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Goldfarb, D. and Liu, S. (1988), An O ( n 3 L )Primal Interior Point Algorithm for Convex Quadratic Programming, Manuscript, Department of Industrial Engineering and Operations Research, Columbia University, New York, NY.Google Scholar
  9. [9]
    Goldman, A. J. and Tucker, A. W. (1956). Theory of linear programming. Linear Inequalities and Related Systems (H. E. Kuhn and A. W. Tucker, eds.), Annals of Mathematical Studies, No. 38, Princeton University Press, Princeton, New Jersey, 53–97.Google Scholar
  10. [10]
    Gonzaga, C.C. (1989), An Algorithm for Solving Linear Programming Problems in O ( n 3 L ) Operations, In Progress in Mathematical Programming Interior Point and Related Methods 1–28, N. Megiddo ed., Springer Verlag, New York.CrossRefGoogle Scholar
  11. [11]
    Gonzaga, C.C. (1990), Polynomial Affine Algorithms for Linear Programming Mathematical Programming 49, 7–21.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Güler, O., Roos, C., Terlaky, T., Vial, J.-Ph. (1992), Interior Point Approach to the Theory of Linear Programming, Cahiers de Recherche No. 1992.3, Faculté des Sciences Economiques et Sociales, Université de Genève.Google Scholar
  13. [13]
    Jansen, B., Roos, C., Terlaky, T. (1992), Interior Point Approach to Postoptimal and Parametric Analysis in Linear Programming, Working paper.Google Scholar
  14. [14]
    Karmarkar, N. (1984). A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Kojima, M., Mizuno, S., and Yoshise, A. (1989). A primal-dual interior point algorithm for linear programming. Progress in Mathematical programming Interior Point and Related Methods (N. Megiddo, ed.), Springer Verlag, New York, 29–48.CrossRefGoogle Scholar
  16. [16]
    Kojima, M., Mizuno, S., Yoshise, A. (1989), A Polynomial Time Algorithm for a Class of Linear Complementarity Problems Mathematical Programming 44, 1–26.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    McLinden, L. (1980). An analogue of Moreau’s proximation theorem, with applications to the nonlinear complementarity problem. Pacific Journal of Mathematics 88, 101–161.MathSciNetzbMATHGoogle Scholar
  18. [18]
    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.zbMATHCrossRefGoogle Scholar
  19. [19]
    Mizuno, S., Todd, M.J. (1989), An O ( n 3 L ) Long Step Path Following Algorithm for a Linear Complementarity Problem, Technical Report, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY.Google Scholar
  20. [20]
    Monteiro, R.D.C. and Adler, I. (1989). Interior path following primal-dual algorithms. Part I: Linear programming. Mathematical Programming 44, 2741.Google Scholar
  21. [21]
    Renegar, J. (1988), A Polynomial-Time Algorithm, Based on Newton’s Method, for Linear Programming Mathematical Programming 40, 59–93.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Rockafellar, R. T. (1970). Convex Analysis .Princeton University Press, Princeton, New Jersey.zbMATHGoogle Scholar
  23. [23]
    Roos, C. (1987), A New, Trajectory Following Polynomial-Time Algorithm for the Linear Programming Problem Journal on Optimization Theory and its Applications 63, 433–458.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Roos, C. and Vial, J.-Ph. (1988), A Polynomial Method of Approximate Centers for Linear Programming Mathematical Programming 54, No. 3, forthcoming.Google Scholar
  25. [25]
    C. Roos and J.Ph. Vial (1990). Long Steps with the Logarithmic Penalty Barrier Function in Linear Programming .In Economic Decision-Making: Games , Economics and Optimization (dedicated to Jacques H. Drèze), edited by J. Gabszevwicz, J.-F. Richard and L. Wolsey, Elsevier Science Publisher B.V., 433–441.Google Scholar
  26. [26]
    Schrijver, A. (1986). Theory of Linear and Integer Programming .John Wiley & Sons, New York.zbMATHGoogle Scholar
  27. [27]
    Sonnevend, Gy. (1985) An “analytic center” for polyhedrons and new classes of global algorithms for linear (smooth convex) programming. In A. Prékopa, J. Szelersân and B. Strazicky, eds. System Modelling and Optimization: Proceedings of the 12th IFIP-Conference Budapest, Hungary September 1985 Vol. 84. of Lecture Notes in Control and Information Sciences Springer Verlag, Berlin, 866–876.Google Scholar
  28. [28]
    Strang, G. (1988). Linear Algebra and its Applications (3rd edition). Harcourt Brace Jovanovich Publishers, San Diego, California.Google Scholar
  29. [29]
    Todd, M.J. and Burrell, B.P. (1986), An Extension of Karmarkar’s Algorithm for Linear Programming Using Dual Variables Algorithmica 1, 409–424.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    Todd, M. J. and Ye, Y. (1990). A centered projective algorithm for linear programming. Mathematics of Operations Research 15, 508–529.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    Vaidya, P.M. (1990), 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.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    Vanderbei, R.J., Meketon, M.S., Freedman, B.A. (1986), A Modification of Karmarkar’s Linear Programming Algorithm Algorithmica 1, 395–407.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    Ye, Y. (1987), Interior Algorithms for Linear, Quadratic and Linearly Constrained Convex Programming, Ph.D. Dissertation, Engeneering-Economic Systems Department, Stanford University, Stanford.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1992

Authors and Affiliations

  • C. Roos
    • 1
  1. 1.Faculty of Technical Mathematics and Informatics DelftUniversity of TechnologyDelftThe Netherlands

Personalised recommendations