An interior point method in Dantzig–Wolfe decomposition

This paper studies the application of interior point methods in Dantzig–Wolfe decomposition. The main idea is to develop strategies for finding useful interior points in the dual of the restricted master problem as an alternative to finding an optimal solution or the analytic center. The method considers points on the central path between the optimal solution and the analytic center, and thus it includes the previous instances as extreme cases. For a given duality gap there exists a unique primal–dual solution on the central path. We use this solution for some choice of the duality gap. The desired duality gap is either kept fixed in all master iterations or it is updated according to some strategy. We test the method on a number of randomly generated problems of different sizes and with different numbers of subproblems. For most problems our method requires fewer master iterations than the classical Dantzig–Wolfe and the analytic center method. This result is especially true for problems requiring many master iterations. In addition to experiments using an interior point method on the master problems, we have also performed some experiments with an interior point method on the subproblems. Instead of finding an optimal solution for the problems we have developed a strategy that selects a feasible solution having a reduced cost below some prescribed level. Our study focuses on comparative experiments.