To solve large-scale traveling salesman problem (TSP) with better performance, this paper proposes an entropy-based dynamic heterogeneous ant colony optimization (EDHACO). The allotropic mechanism and the heterogeneous colonies model are proposed to balance the convergence and the solution diversity. First, entropy is used to measure the diversity, and the entropy-based allotropic mechanism with three communication strategies can improve the adaptability of EDHACO. Then, the heterogeneous colonies with complementary advantages are proposed to balance the convergence speed and the diversity of the algorithm. Besides, two operators are proposed to improve the performance of the algorithm. The adaptive 3-opt operator can be used to accelerate the convergence, and the dynamic-pheromone-reset operator can be introduced to avoid trapping in a local optimum. Finally, EDHACO is applied to solve TSPs, and the experimental results suggest that it has better performance with higher stability and precision in TSP instances, especially in the large-scale TSP instances. INDEX TERMS Entropy, allotropic mechanism, dynamic pheromone operator, adaptive 3-opt operator, travel salesman problem.
In this paper, we investigate optimal linear approximations (napproximation numbers ) of the embeddings from the Sobolev spaces H r (r > 0) for various equivalent norms and the Gevrey type spaces G α,β (α, β > 0) on the sphere S d and on the ball B d , where the approximation error is measured in the L 2 -norm. We obtain preasymptotics, asymptotics, and strong equivalences of the above approximation numbers as a function in n and the dimension d. We emphasis that all equivalence constants in the above preasymptotics and asymptotics are independent of the dimension d and n. As a consequence we obtain that for the absolute error criterion the approximation problems I d : H r → L 2 are weakly tractable if and only if r > 1, not uniformly weakly tractable, and do not suffer from the curse of dimensionality. We also prove that for any α, β > 0, the approximation problems I d : G α,β → L 2 are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if α ≥ 1. r/d exist, having the same value for various norms. We also prove that for 0 < α < 1,, lim n→∞ e βγn α/d a n (I d : G α,β (S d ) → L 2 (S d )) = 1.
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