Two-stage updating pheromone for invariant ant colony optimization algorithm

Ant colony optimization (ACO) is a metaheuristic approach for combinatorial optimization problems. With the introduction of hypercube framework, invariance property of ACO algorithms draws more attention. In this paper, we propose a novel two-stage updating pheromone for invariant ant colony optimization (TSIACO) algorithm. Compared with standard ACO algorithms, TSIACO algorithm uses solution order other than solution itself as independent variable for quality function. In addition, the pheromone trail is updated with two stages: in one stage, the first r iterative optimal solutions are employed to enhance search capability, and in another stage, only optimal solution is used to accelerate the speed of convergence. And besides, the pheromone value is limited to an interval. We prove that TSIACO not only has the property of linear transformational invariance but also has translational invariance. We also prove that the pheromone trail can limit to the interval (0,1]. Computational results on the traveling salesman problem show the effectiveness of TSIACO algorithm.