Gradient Methods for Binary Integer Programming

Integer Programming (IP) is a common problem class where decision variables represent indivisibility or yes/no decisions. IP can also be interpreted as discrete optimizations which are extensively used in the areas of routing decisions, fleet assignment, and aircraft/aircrew scheduling. Solving IP models is much harder than solving Linear Programming (LP) models because of their intrinsically combinatorial nature. In addition, many “real-world” problems have a huge problem size that makes solving the related IP-based model time-consuming. In this paper, we propose a gradient method for binary IP that enable large problems to be solved in a short amount of time. The algorithm is partitioned into two phases. In phase 1 of the algorithm, integrality constraints are modified and the original problem is solved by the LP algorithm. In phase 2, all integrality constraints are modified again and added back. The Lagrange multipliers are then introduced to form a Lagrange dual problem. We then use the gradient method to quickly search for the optimal or nearly optimal solution. After developing the algorithm, we show the effectiveness of this approach by applying it to various kinds of binary IP problems with discussions about the reduction in computation time and the gap between objective functions.