On the worst case complexity of potential reduction algorithms for linear programming

There are several classes of interior point algorithms that solve linear programming problems in O(Vin L) iterations, but it is not known whether this bound is tight for any interior point algorithm. Among interior point algorithms, several potential reduction algorithms combine both theoretical (O(+/E L) iterations) and practical efficiency as they allow the flexibility of line searches in the potential function and thus can lead to practical implementations. It is a significant open question whether interior point algorithms can lead to better complexity bounds. In the present paper we give some negative answers to this question for the class of potential reduction algorithms. We show that, without line searches in the potential function, the bound O(v/iL) is tight for several potential reduction algorithms, i.e., there is a class of examples, in which the algorithms need at least Q(v/i L) iterations to find an optimal solution. In addition, we show that for a class of potential functions, even if we allow line searches in the potential function, the bounds are still tight. We note that it is the first time that tight bounds are obtained for any interior point algorithm. III