Abstract
We investigate the decrease in potential at an iteration of Karmarkar's projective method for linear programming. For a fixed step length parameterα (so that we must have 0 <α ≤ 1) the best possible guaranteeδ n (α) inn dimensional space is essentially ln 2 ≃ 0.69; and to achieve this we must takeα about 1. Indeed we show the precise result thatδ n (α) equals ln(1 +α)-ln(1 −α/(n − 1)) forn sufficiently large. If we choose an optimal step length at each iteration then this guarantee increases only to aboutδ * ≃ 0.72. We also shed some light on the remarkable empirical observation that the number of iterations required seems scarcely to grow with the size of the problem.
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Mcdiarmid, C. On the improvement per iteration in Karmarkar's algorithm for linear programming. Mathematical Programming 46, 299–320 (1990). https://doi.org/10.1007/BF01585747
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DOI: https://doi.org/10.1007/BF01585747