The facet ascending algorithm for integer programming problems

Many practical large-scale optimization problems, such as scheduling a manufacturing system, can be modeled as integer programming problems. Because of their combinatorial nature, these problems are often very difficult to solve optimally, especially within a limited amount of time. Therefore, near-optimal solutions are often sought. Lagrangian relaxation is an effective method for decomposing a difficult problem into subproblems that are much easier to solve. A major part of this method is to optimize the dual function of the integer programming problem. Since the dual function is nondifferentiable, the subgradient method is frequently used as a method for maximizing (for a primal minimization problem) the dual function. However, this method can exhibit slow convergence due to iterations zigzagging across a set of nondifferentiable points. The new algorithm presented in this paper exploits the polyhedral concave nature of the dual function by ascending facets along nondifferentiable points, thus avoiding the zigzagging behaviour of the subgradient method. The algorithm is tested on a nonlinear integer programming problem for scheduling a simple manufacturing system. The computational results show this algorithm is a significant improvement over the subgradient method.<>