On the complexity of linear programming under finite precision arithmetic

In this paper we study the complexity of solving linear programs in finite precision arithmetic. This is the normal setup in scientific computation, as digital computers work in finite precision. We analyze two aspects of the complexity: one is the number of arithmetic operations required to solve the problem approximately, and the other is the working precision required to carry out some critical computations safely. We show how the ‘conditioning’ of the problem instance affects the working precision required and the computational requirements of a classical logarithmic barrier algorithm to approximate the optimal value of the problem within a given tolerance. Our results show that these complexity measures depend linearly on the logarithm of a certain condition measure. We carry out the analysis by looking at how well Newton's Method can follow the central trajectory of the feasible set, and computing error bounds in terms of the condition measure. These results can be interpreted as a theoretical indication of ‘good’ numerical behavior of the logarithmic barrier method, in the sense that a problem instance ‘twice as hard’ as the other from the numerical point of view, requires only at most twice as much precision to be solved.