Anticipated Behavior of Long-Step Algorithms for Linear Programming.

We provide a probabilistic analysis of the second order term that arises in path-following algorithms for linear programming. We use this result to show that two such methods, algorithms generating a sequence of points in a neighborhood of the central path and in its relaxation , require a worst-case number of iterations that is O(nL) and an anticipated number of iterations that is O(log(n)L). The second neighborhood spreads almost all over the feasible region so that the generated points are close to the boundary rather than the central path. We also 0 propose a potential reduction algorithm which requires the same order of number of iterations as the path-following algorithms.

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