The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard combinatorial optimization problems. In order to deal with large TSP instances, several heuristics and metaheuristics have been devised. In this paper, a novel memetic scheme that incorporates a new diversity-based replacement strategy is proposed and applied to the largest instances of the TSPLIB benchmark. The novelty of our method is that it combines the idea of transforming a single-objective problem into a multi-objective one, by considering diversity as an explicit objective, with the idea of adapting the balance induced between exploration and exploitation to the various optimization stages. In addition, the intensification capabilities of the individual learning method incorporated in the memetic scheme are also adapted by taking into account the stopping criterion. Computational results show the clear superiority of our scheme when compared against state-of-the-art schemes. To our knowledge, our proposal is the first evolutionary scheme that readily solves an instance with more than 30,000 cities to optimality.