Using logical surrogate information in Lagrangean relaxation: An application to symmetric traveling salesman problems

The Traveling Salesman Problem (TSP) is a classical Combinatorial Optimization problem, which has been intensively studied. The Lagrangean relaxation was first applied to the TSP in 1970. The Lagrangean relaxation limit approximates what is known today as HK (Held and Karp) bound, a very good bound (less than 1% from optimal) for a large class of symmetric instances. It became a reference bound for new heuristics, mainly for the very large scale instances, where the use of exact methods is prohibitive. A known problem for the Lagrangean relaxation application is the definition of a convenient step size control in subgradient like methods. Even preserving theoretical convergence properties, a wrong defined control can affect the performance and increase computational times. We show in this work how to accelerate a classical subgradient method while conserving good approximations to the HK bounds. The surrogate and Lagrangean relaxation are combined using the local information of the relaxed constraints. It results in a one-dimensional search that corrects the possibly wrong step size and is independent of the used step size control. Comparing with the ordinary subgradient method, and beginning with the same initial multiplier, the computational times are almost twice as fast for medium instances and greatly improved for some large scale TSPLIB instances.