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A Dantzig-Wolfe-Like Variant of Karmarkar's Interior-Point Linear Programming Algorithm

Author

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  • Michael J. Todd

    (Cornell University, Ithaca, New York)

Abstract

We show that a variant of Karmarkar's projective algorithm for linear programming can be viewed as following the approach of Dantzig-Wolfe decomposition. At each iteration, the current primal feasible solution generates prices which are used to form a simple subproblem. The solution to the subproblem is then incorporated into the current feasible solution. With a suitable choice of stepsize a constant reduction in potential function is achieved at each iteration. We also use our analysis to motivate a new primal simplex pivot rule that is closely related to rules used by E. Klotz and L. Schrage.

Suggested Citation

  • Michael J. Todd, 1990. "A Dantzig-Wolfe-Like Variant of Karmarkar's Interior-Point Linear Programming Algorithm," Operations Research, INFORMS, vol. 38(6), pages 1006-1018, December.
    • Handle: RePEc:inm:oropre:v:38:y:1990:i:6:p:1006-1018
    DOI: 10.1287/opre.38.6.1006
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    Cited by:

    1. Holder, A.G. & Sturm, J.F. & Zhang, S., 1998. "Analytic central path, sensitivity analysis and parametric linear programming," Econometric Institute Research Papers EI 9801, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    2. Zhang, S., 1998. "Global error bounds for convex conic problems," Econometric Institute Research Papers EI 9830, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    3. A.G. Holder & J.F. Sturm & S. Zhang, 1998. "Analytic Central Path, Sensitivity Analysis and Parametric Linear Programming," Tinbergen Institute Discussion Papers 98-003/4, Tinbergen Institute.
    4. J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.
    5. Dennis Cheung & Felipe Cucker & Javier Peña, 2003. "Unifying Condition Numbers for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 609-624, November.
    6. Li-Zhi Liao, 2014. "A Study of the Dual Affine Scaling Continuous Trajectories for Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 548-568, November.

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