Lagrangian Heuristics to Linear Ordering

Two heuristics for the Linear Ordering Problem are investigated in this paper. heuristics are embedded within a Lagrangian Relaxation framework and are initiated with a construction phase. In this process, some Lagrangian (dual) information is used as an input to guide the construction of initial Linear Orderings. Solutions thus obtained are then submitted to local improvement in an overall procedure that is repeated for every Subgradient Optimization iteration. On the Lagrangian relaxation side of the algorithm, Relax and Cut is used instead of a straightforward implementation of the Subgradient Method. The reason being the exceedingly large number of inequalities that must be relaxed to generate (Lagrangian) dual bounds. The proposed heuristics are tested for instances from the literature and also for some new hard to solve exactly ones. In particular, optimal solutions are obtained for all 79 instances taken from the literature. As a by product, optimality is also proven for 72 of them.