Search directions for interior linear-programming methods

Since Karmarkar published his algorithm for linear programming, several different interior directions have been proposed and much effort was spent on the problem transformations needed to apply these new techniques. This paper examines several search directions in a common framework that does not need any problem transformation. These directions prove to be combinations of two problem-dependent vectors, and can all be improved by a bidirectional search procedure.We conclude that there are essentially two polynomial algorithms: Karmarkar's method and the algorithm that follows a central trajectory, and they differ only in a choice of parameters (respectively lower bound and penalty multiplier).

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