The design of algorithms in swarm and evolutionary computation has shown that to avoid the premature con- vergence problem during an optimization process of multimodal functions, the algorithms must provide a proper equilibrium between global search (exploration) and local search (exploitation) at any stage of the search. Several strategies have been investigated, including the use of heavy-tailed distributions for sampling the search space of the problem. It has been conjectured that heavy-tailed distributions can increase the chances of a particle (or in- dividual in a population) to move away from a local optimum. The general idea is to allow exploration in regions far away from the current position. Following this idea, two works in evolutionary computation using heavy-tailed distributions can be highlighted, namely the study developed by Yao, Liu & Lin (1999) and also the study devel- oped Lee & Yao (2004). Yao, Liu & Lin (1999) introduced a fast evolutionary programming (FEP) which uses mutations based on the Cauchy distribution instead of mutations based on the Gaussian distribution, which is used
in the classical evolutionary programming (CEP). Experimental results show that FEP performs much better than CEP for multimodal functions with many local minima, while it remains comparable in performance to CEP for unimodal functions and multimodal functions with only a few local minima. This work report the fact that Cauchy mutation performs better when the current search point is far away from the global minimum, while Gaussian mutation is better at finding a local minimum in a good region. Lee & Yao (2004) introduced a evolutionary programming (EP) with mutations based on the L´evy distribution. This distribution is symmetrical with respect to zero and has two parameters α and γ, where γ is the scaling factor satisfying γ > 0 and α controls the shape of the distribution satisfying 0 < α ≤ 2. Its density resembles a bell curve, such as a Gaussian density, but with heavier tails (α can be used to control the heaviness of the tails). The L´evy distribution includes some models as special cases: the Cauchy distribution when α = 1 and the normal distribution when α = 2. An algorithm for generating L´evy random numbers was introduced in Mantegna (1994). The work developed by Lee & Yao (2004) reports empirical evidences that the performance of the EP with L´evy mutation was better than that of the CEP for multimodal functions with many local minima since that L´evy mutation is more general and flexible than Cauchy and Gaussian mutations.
BBPSO has shown potential for solving optimization problems defined on continuous search spaces. However, it suffers from the premature convergence problem when solving multimodal problems. In order to address this drawback and improve the performance of the original algorithm, some strategies have been developed. New variants of BBPSO have been proposed (Richer & Blackwell, 2006; Krohling & Mendel, 2009; Hsieh & Lee, 2010; Blackwell, 2012; Liu, Ding & Wang, 2014; Campos, Krohling & Enriquez, 2014) and some of these variants are discussed in Sections B.7 and B.8.
Richer & Blackwell (2006) investigated the effectiveness of using the L´evy distribution in BBPSO. Based on a series of trials, this work reports that the L´evy BBPSO with α = 1.4 reproduces the behaviour of the standard PSO (Bratton & Kennedy, 2007). These results support the conjecture that heavy-tailed distributions to update the position of a particle provide an increase in its ability to move away from a local minimum, improving at the same time the balance between exploration and exploitation.
Krohling & Mendel (2009) introduced a BBPSO with a particular jump strategy, when no improvement on the value of the objective function is observed. This jump strategy was implemented based on the Gaussian or Cauchy distributions. The algorithm was tested on a well-known suite of benchmark multimodal functions and the results were compared with those obtained by the BBPSO algorithm. Simulation results have shown that the BBPSO algorithm with jump strategy has performed well in all functions investigated. Krohling & Mendel (2009) also pointed that the improved performance was due to a successful number of Gaussian or Cauchy jumps, with a performance slightly better for the case of Cauchy jumps. This result is essentially compatible with the conjecture that distributions with heavy tails increase the chances of a particle to escape from local minima, exploring the search space without losing exploitation in promising regions.
Blackwell (2012) formulated the dynamic update rule of the PSO as a second-order stochastic difference equa- tion. This formulation was used to derive general expressions for search focus, search spread, and swarm stability at stagnation. The results were applied to three swarm algorithms: the standard PSO, PSO with discrete recombi- nation, and BBPSO. Blackwell (2012) proposed a generalized BBPSO (GBBPSO) such that the search focus and the search spread can each one be chosen from the global or local neighborhoods. The position of a particle is updated as xτ +1 k = µµµ τ +1 k + αδδδ τ +1 k z (B.61)
for all k and τ ≥ 0, where α > 02and
• µµµτ +1k = BEST(pτl : l ∈Nk) or BEST(pτk: k = 1, . . . , K)
• δδδτ +1
k = |pτk− nτk| or |pkτ− gτ| or |pτk+1− pτk−1| (mod K)
• z ∼ N(0, I) = (N(0, 1),. . ., N(0, 1))iid 0.
The theoretical analysis, presented by Blackwell (2012), predicts a no-collapse condition when α > αc= 0.65.
Collapse is a swarm pathology, where the particles approach each other faster than the swarm as a whole approaches a local minimum. As a result, the convergence toward a limit point becomes exponentially slow and the swarm stagnates. Another important result is that the fastest rate of convergence of the GBBPSO occurs at the critical 2Note that in the GBBPSO algorithm, the parameter α has a conceptual meaning completely different from that the parameter α has in the
value αc. Experimental results confirm that GBBPSO situated at the edge of collapse is comparable to the standard
PSO and PSO with discrete recombination. The main loop of GBBPSO is shown in Algorithm 14. Algorithm 14 GBBPSO and GBBPSOwJ.
Input: α > 0 1: {GBBPSO} 2: loop 3: for k ∈ {1, . . . , K} do 4: µµµk← BEST(pl: l ∈Nk) 5: δδδk← |pk− nk| 6: for d ∈ {1, . . . , D} do 7: z∼ N(0, 1) 8: xkd← µkd+ αδkd· z 9: end for 10: pk← BEST(xk, pk) 11: end for 12: for k ∈ {1, . . . , K} do 13: nk← BEST(pl: l ∈Nk) 14: end for 15: end loop Input: α > 0 and 0 ≤ pJ< 1 1: {GBBPSOwJ} 2: loop 3: for k ∈ {1, . . . , K} do 4: µµµk← BEST(pl: l ∈Nk) 5: δδδk← |pk− nk| 6: for d ∈ {1, . . . , D} do 7: r∼ Unif(0, 1) 8: if r < pJthen 9: xkd∼ Unif(Ld,Ud) 10: else 11: z∼ N(0, 1) 12: xkd← µkd+ αδkd· z 13: end if 14: end for 15: pk← BEST(xk, pk) 16: end for 17: for k ∈ {1, . . . , K} do 18: nk← BEST(pl: l ∈Nk) 19: end for 20: end loop
Additional results indicated that the performance of the proposed algorithm can be still further improved with the use of an adaptive distribution with heavy tails. In fact, Blackwell (2012) also proposed a GBBPSO with jumps (GBBPSOwJ) following a different strategy from that which was considered by Krohling & Mendel (2009). GBBPSOwJ can be seen as the GBBPSO combined with a probabilistic jumping mechanism: a particle may jump uniformly in any dimension with probability pJ. This strategy can be seen as a partial re-initialization (since, in
general, not every component undergoes a jump) or, alternatively, as a mechanism to fatten the tails of the normal distribution that generates new positions in the search space, allowing search in areas where the tails of the normal distribution are thin. This adaptive distribution allows exploration throughout the search volume at any stage of the optimization. The main loop of GBBPSOwJ is shown in Algorithm 14.
Hsieh & Lee (2010) also introduced a modified BBPSO. Empirical studies, considering a well-known set of test functions, have showed that the proposed algorithm is a competitive optimizer due to its good performance and fast convergence rate. The BBPSO proposed by Hsieh & Lee (2010) shows some similar characteristics to BBPSO proposed by Blackwell (2012), with the former having an additional parameter to focus the search.