Linear Programming pp 361-381 | Cite as

# The Homogeneous Self-Dual Method

Chapter

## Abstract

In Chapter 18, we described and analyzed an interior-point method called the path-following algorithm. This algorithm is essentially what one implements in practice but as we saw in the section on convergence analysis, it is not easy (and perhaps not possible) to give a complete proof that the method converges to an optimal solution. If convergence were completely established, the question would still remain as to how fast is the convergence. In this chapter, we shall present a similar algorithm for which a complete convergence analysis can be given.

## Keywords

Step Length Dual Problem Linear Programming Problem Simplex Method Interior Point Method
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## References

- Tucker, A. (1956), ‘Dual systems of homogeneous linear equations’,
*Annals of Mathematics Studies***38**, 3–18.MATHGoogle Scholar - Mizuno, S., Todd, M. & Ye, Y. (1993), ‘On adaptive-step primal-dual interior-point algorithms for linear programming’,
*Mathematics of Operations Research***18**, 964– 981.MATHCrossRefMathSciNetGoogle Scholar - Ye, Y., Todd, M. & Mizuno, S. (1994), ‘An
*o(√nl)*-iteration homogeneous and selfdual linear programming algorithm’,*Mathematics of Operations Research***19**, 53– 67.MATHCrossRefMathSciNetGoogle Scholar - Xu, X., Hung, P. & Ye, Y. (1993), A simplified homogeneous and self-dual linear programming algorithm and its implementation, Technical report, College of Business Administration, University of Iowa. To appear in Annals of Operations Research.Google Scholar
- Mehrotra, S. (1992), ‘On the implementation of a (primal-dual) interior point method’,
*SIAM Journal on Optimization***2**, 575–601.MATHCrossRefMathSciNetGoogle Scholar - Mehrotra, S. (1989), Higher order methods and their performance, Technical Report TR 90-16R1, Department of Ind. Eng. and Mgmt. Sci., Northwestern University, Evanston, IL. Revised July, 1991.Google Scholar
- Adler, I., Karmarkar, N., Resende, M. & Veiga, G. (1989), ‘An implementation of Karmarkar’s algorithm for linear programming’,
*Mathematical Programming***44**, 297– 335.MATHCrossRefMathSciNetGoogle Scholar

## Copyright information

© Robert J.Vanderbei 2008