Adaptive Use of Iterative Methods in Interior Point Methods for Linear Programming

In this work we devise eecient algorithms for nding the search directions for interior point methods applied to linear programming problems. There are two innovations. The rst is the use of updating of preconditioners computed for previous barrier parameters. The second is an adaptive automated procedure for determining whether to use a direct or iterative solver, whether to reinitialize or update the preconditioner, and how many updates to apply. These decisions are based on predictions of the cost of using the diierent solvers to determine the next search direction, given costs in determining earlier directions. These ideas are tested by applying a modiied version of the OB1-R code of Lustig, Marsten, and Shanno to a variety of problems from the NETLIB and other collections. If a direct method is appropriate for the problem, then our procedure chooses it, but when an iterative procedure is helpful, substantial gains in eeciency can be obtained.