Many classes of combinatorial and mixed integer optimization problems are attacked with decomposition methods. One technique is to perform subgradient optimization on a Lagrangean dual problem: another is to perform column generation on a Dantzig-Wolfe problem, or equivalently, cut generation on its linear programming dual. These techniques both have advantages and disadvantages. In this paper these techniques are combined into a two-phase method, with the purpose of overcoming drawbacks of the techniques.
The two-phase method consists of a prediction phase and a solution phase. In the prediction phase, subgradient optimization is performed: subgradients found are stored and used as starting columns for the solution phase. (Optionally, non-binding restricitions can be predicted based on information from the prediction phase.) The columns found are used to construct a Dantzig/Wolfe master problem. In the solution phase, column generation is performed if needed. A solid theoretical bases for the two-phase method is presented which includes strong asymptotical results. ln practise, truncated usage must be performed: practical guidelines are given in the paper.
The two-phase method is tested on two applications: a multicommodity network flow problem and a convexified version of the traveling salesman subtour problem. Two categories of numerical experiments are presented. ln the first category, various aspects of truncated usage of the theory are illustrated. In the second category, the two-phase method is tested on a relatively large number of test problems.
An overall conclusion of our work is that the two-phase method can perform significantly better, in terms of CPU-times, compared to a (stabilized) Dantzig-Wolfe algorithm. ln general, whenever the subproblems are computationaly inexpensive, compared to the Dantzig-Wolfe master programs, the two-phase method might be an interesting alternative to use instead of pure Dantzig-Wolfe.