# Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems
## 3 Search Patterns in Metaheuristics  
The generation of efficient search patterns for the correct exploration of a fitness landscape could be complicated, particularly in the presence of ruggedness and multiple local optima. Recently, several new metaheuristic schemes have been introduced in the literature. Although all these approaches consider very different phenomena as metaphors, the search patterns, used to explore the search space, are very similar. A search pattern is a set of movements produced by a rule or model in order to examine promising solutions from the search space.  
Exploration and exploitation correspond to the most important characteristics of a search pattern. Exploration refers to the ability of a search pattern to examine a set of solutions spread in distinct areas of the search space. On the other hand, exploitation represents the capacity of a search pattern to improve the accuracy of the existent solutions through a local examination. The combination of both mechanisms in a search pattern is crucial for attaining success when solving a particular optimization problem.  
To solve the optimization formulation, from a metaheuristic point of view, a population of \(\mathbf{P}^{k}\)\(\left(\left\{\mathbf{x}_{1}^{k},\ldots,\mathbf{x}_{N}^{k}\right\}\right)\) of \(N\) candidate solutions (individuals) evolve from an initial point (\(k=1\)) to a \(Maxgen\) number of generations (\(k=Maxgen\)). In the population, each individual \(\mathbf{x}_{i}^{k}\)\((i\in[1,\ldots,N])\) corresponds to a \(d\)-dimensional element \(\left\{x_{i,1}^{k},\ldots,x_{i,d}^{k}\right\}\), which symbolizes the decision variables involved by the optimization problem. At each generation, search patterns are applied over the individuals of the population \(\mathbf{P}^{k}\) to produce the new population \(\mathbf{P}^{k+1}\). The quality of each individual \(\mathbf{x}_{i}^{k}\) is evaluated in terms of its solution regarding the objective function \(J\left(\mathbf{x}_{i}^{k}\right)\) whose result represents the fitness value of \(\mathbf{x}_{i}^{k}\). As the metaheuristic method evolves, the best current individual \(\mathbf{b}\)\(\left\{b_{1},\ldots,b_{d}\right\}\) is maintained since \(\mathbf{b}\) represents the best available solution seen so-far.  
In general, a search pattern is applied to each individual \(\mathbf{x}_{i}^{k}\) using the best element \(\mathbf{b}\) as a reference. Then, following a particular model, a set of movements are produced to modify the position of \(\mathbf{x}_{i}^{k}\) until the location of \(\mathbf{b}\) has been reached. The idea behind this mechanism is to examine solutions in the trajectory from \(\mathbf{x}_{i}^{k}\) to \(\mathbf{b}\) with the objective to find a better solution than the current \(\mathbf{b}\). Search patterns differ in the model employed to produce the trajectories \(\mathbf{x}_{i}^{k}\) from to \(\mathbf{b}\).  
Two of the most popular search models are attraction and spiral trajectories. The attraction model generates attraction movements from \(\mathbf{x}_{i}^{k}\) to \(\mathbf{b}\). The attraction model is used extensively by several metaheuristic methods such as PSO [10; 11; 12], FA [13; 14], CS [15], BA [16], GSA [24], EM [27] and DE [6]. On the other hand, the spiral model produces a spiral trajectory that encircles the best element \(\mathbf{b}\). The spiral model is employed by the recently published WOA and GWO schemes. Trajectories produced by the attraction, and spiral models are visualized in Figure 3a,b, respectively.