Arguments for ACO's Success

Very little theory is available to explain the reasons underlying ACO’s success. A population–based ACO (P-ACO) variant is used to explain the reasons of elitist ACO’s success in the TSP, given a globally convex structure of the solution space. 1 Reasons Underlying ACO’s Success For this work a TSP tour is denoted as rx, the optimal tour as r∗ and a population of m tours as P = {Pi}. Distance δ(rx, ry) is defined as the number of cities n minus the number of common arcs between tours rx and ry. Inspired in [1], Fig. 1 (a) presents the length of a tour l(rx) as a function of its distance to the optimal solution δ(rx, r∗) for the whole space S of a randomly chosen TSP with 8 cities. Fig. 1 (b) shows the length of rx ∈ S as a function of its mean distance to a population δ(P, rx) = 1 m ∑m i=1 δ(Pi, rx) of randomly chosen good solutions for the same problem. As previously found for different TSP instances [1], a positive correlation is observed. Consequently, the TSP solution space has a globally convex structure for all tested instances [1]. Fig. 1. Distance of the 2,520 solutions of the randomly chosen TSP with 8 cities To understand the typical behavior of ACO, the n–dimensional TSP search space is simplified to two dimensions for a geometrical vision in Fig. 2. A population P1 = {P1i} of good solutions uniformly distributed is assumed in Fig. 2. Considering that the proposed variant of P-ACO [2] (called Omicron ACO or K. Deb et al. (Eds.): GECCO 2004, LNCS 3102, pp. 259–260, 2004. c © Springer-Verlag Berlin Heidelberg 2004 260 O. Gómez and B. Barán Fig. 2. Simplified vision of OA behavior OA) gives more pheromones to the good solutions P1i already found, this can be seen as a search made close to each P1i. Thus, OA concentrates the search of new solutions in a central zone of P1, denoted as ΩP1, which is the zone close to all P1i. Then OA typically replaces the worst solution of P1 (P1worst) by a new solution Pnew of smaller length. A new population P2 is created including Pnew. This is shown in Fig. 2 with a dotted line arrow. As a consequence, it is expected that δ(P2, r∗) < δ(P1, r∗) because there is a positive correlation between l(rx) and δ(rx, r∗). Similarly, δ(P, Pnew) < δ(P, Pworst) because there is a positive correlation between l(rx) and δ(P, rx), therefore δ(P2) < δ(P1) (where δ(P ) = 2 m(m−1) ∑m−1 i=1 ∑m j=i+1 δ(Pi, Pj) is the mean distance of P ), i.e. it is expected that the subspace where the search of potential solutions is concentrated decreases. OA performs this procedure repeatedly to decrease the search zone where promising solutions are located, as seen in Fig. 1 (b). Considering population Pz = {Pzi} for z >> 2, Fig. 2 shows how ΩPz has decreased considerably as a consequence of the globally convex structure of the TSP solution space. 2 Conclusions OA concentrates the search in a central zone ΩP of its population P . In globally convex problems, good solutions are usually found in this region; therefore, OA concentrates its search in a promising subspace. Every time a good solution is found, it enters the population reducing the promising search zone iteratively.