An interior point method in Dantzig–Wolfe decomposition

Abstract

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. Scope and purpose Dantzig–Wolfe decomposition is a well established discipline in linear programming, and most textbooks in the area include a chapter on this topic. It is a procedure of fundamental importance for the development of the so-called column generation technique for the solution of large scale problems in optimization as well as for the development of decentralized planning systems in economics. Most presentations and applications of the procedure are done within the context of the simplex method. However with the active development of interior point methods there is an increasing interest also to use those methods in decomposition including the Dantzig–Wolfe decomposition. The present manuscript is a part of this development. Some methods are here proposed linking Dantzig–Wolfe decomposition to some of the decomposition procedures using interior point methods. Comparative experiments are performed pointing out places, where it is advantageous to use an interior point method.