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An assumption‐free convergence analysis for a perturbation of the scaling algorithm for linear programs, with application to the L1 estimation problem

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  • Hanif D. Sherali
  • Bradley O. Skarpness
  • Buyong Kim

Abstract

This article is concerned with the scaling variant of Karmarkar's algorithm for linear programming problems. Several researchers have presented convergence analyses for this algorithm under various nondegeneracy types of assumptions, or under assumptions regarding the nature of the sequence of iterates generated by the algorithm. By employing a slight perturbation of the algorithm, which is computationally imperceptible, we are able to prove without using any special assumptions that the algorithm converges finitely to an ε‐optimal solution for any chosen ε > 0, from which it can be (polynomically) rounded to an optimum, for ε > 0 small enough. The logarithmic barrier function is used as a construct for this analysis. A rounding scheme which produces an optimal extreme point solution is also suggested. Besides the non‐negatively constrained case, we also present a convergence analysis for the case of bounded variables. An application in statistics to the L1 estimation problem and related computational results are presented.

Suggested Citation

  • Hanif D. Sherali & Bradley O. Skarpness & Buyong Kim, 1988. "An assumption‐free convergence analysis for a perturbation of the scaling algorithm for linear programs, with application to the L1 estimation problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(5), pages 473-492, October.
    • Handle: RePEc:wly:navres:v:35:y:1988:i:5:p:473-492
    DOI: 10.1002/1520-6750(198810)35:53.0.CO;2-#
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    References listed on IDEAS

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    1. Kortanek, K. O. & Shi, M., 1987. "Convergence results and numerical experiments on a linear programming hybrid algorithm," European Journal of Operational Research, Elsevier, vol. 32(1), pages 47-61, October.
    2. Hanif D. Sherali, 1987. "Algorithmic insights and a convergence analysis for a Karmarkar‐type of algorithm for linear programming problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(3), pages 399-416, June.
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    Cited by:

    1. Sherali, Hanif D. & Arora, Namita & Hobeika, Antoine G., 1997. "Parameter optimization methods for estimating dynamic origin-destination trip-tables," Transportation Research Part B: Methodological, Elsevier, vol. 31(2), pages 141-157, April.

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