A Path-Following Method

  • Robert J. Vanderbei
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 114)


In this chapter, we define an interior-point method for linear programming that is called a path-following method. Recall that for the simplex method we required a two-phase solution procedure. The path-following method is a one-phase method. This means that the method can begin from a point that is neither primal nor dual feasible and it will proceed from there directly to the optimal solution.


Simplex Method Linear Complementarity Problem Central Path Step Direction Primal Feasibility 
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  1. Kojima, M., Mizuno, S. & Yoshise, A. (1989), A primal-dual interior point algorithm for linear programming, in N. Megiddo, ed., ‘Progress in Mathematical Programming’, Springer-Verlag, New York, pp. 29–47.Google Scholar
  2. Lustig, I. (1990), ‘Feasibility issues in a primal-dual interior-point method for linear programming’, Mathematical Programming 49(2), 145–162.MATHCrossRefMathSciNetGoogle Scholar
  3. Khachian, L. (1979), ‘A polynomial algorithm in linear programming’, Doklady Academiia Nauk SSSR 244, 191–194. In Russian. English Translation: Soviet Mathematics Doklady 20: 191-194.Google Scholar
  4. Karmarkar, N. (1984), ‘A new polynomial time algorithm for linear programming’, Combinatorica 4, 373–395.MATHCrossRefMathSciNetGoogle Scholar
  5. Monteiro, R. & Adler, I. (1989), ‘Interior path following primal-dual algorithms: Part i: Linear programming’, Mathematical Programming 44, 27–41.MATHCrossRefMathSciNetGoogle Scholar
  6. Todd, M. (1995), Potential-reduction methods in mathematical programming, Technical Report 1112, SORIE, Cornell University, Ithaca, Ithaca.Google Scholar
  7. Anstreicher, K. (1996), Potential Reduction Algorithms, Technical report, Department of Management Sciences, University of Iowa.Google Scholar
  8. Wright, S. (1996), Primal-Dual Interior-Point Methods, SIAM, Philadelphia, USA.Google Scholar
  9. Nazareth, J. (1986), ‘Homotopy techniques in linear programming’, Algorithmica 1, 529–535.MATHCrossRefMathSciNetGoogle Scholar
  10. Nazareth, J. (1996), ‘The implementation of linear programming algorithms based on homotopies’, Algorithmica 15, 332–350.MATHCrossRefMathSciNetGoogle Scholar
  11. Carpenter, T., Lustig, I., Mulvey, J. & Shanno, D. (1993), ‘Higher order predictorcorrector interior point methods with application to quadratic objectives’, SIAM Journal on Optimization 3, 696–725.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Robert J.Vanderbei 2008

Authors and Affiliations

  • Robert J. Vanderbei
    • 1
  1. 1.Dept. of Operations Research and Financial EngineeringPrinceton UniversityNew JerseyUSA

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