An improved lower bound for the asymmetric traveling salesman problem based on the assignment problem

In this article we describe how to compute a lower bound for the asymmetric traveling salesman problem that dominates the bound that comes from the assignment relaxation, through the solving of a sequence of assignment problems. The algorithm that we propose is a first-order method based on the exponential penalty function. Directions of movement are derived from a disjunctive relaxation that we proposed as being one of two possible classes, one based on cycles, the other based on cliques.