# K-Means Clustering Algorithm Based on Chaotic Adaptive Artificial Bee Colony
## 3 Chaotic Adaptive Artificial Bee Colony (CAABC) for Clustering
### New Position Update Rules  
#### 3.3.1 Arregate-Dispeise Algorithm  
To improve convergence speed and accuracy, aggregate-disperse algorithm is introduced here. On the basis of the simplex method, we propose an \"aggregate-disperse operation\" as a guiding strategy for the iteration. According to the relationship among the global optimal solution, elite solution and ordinary individual, the search range and step length change adaptively. * The Simplex Method The simplex method is a traditional optimization method which uses the iterative transformation of the vertex of the geometric graph to approach the optimal value gradually. Take a dual function as example. Take three points \(X_{1}\), \(X_{2}\), \(X_{3}\), which are not collinear, as the vertices to form a triangle. Calculate the function value \(f(X_{1})\), \(f(X_{2})\), \(f(X_{3})\) and compare them to each other. Calculate the function value \(f(X_{1})\), \(f(X_{2})\), \(f(X_{3})\) and compare them to each other. (1) \(f(X_{1})>f(X_{2})>f(X_{3})\) means \(X_{1}\) is the worst solution, and \(X_{3}\) is the best one. The algorithm should search in the opposite direction to find the minimum. \(X_{4}=(X_{2}+X_{3})/2\) is the midpoint of \(X_{2}X_{3}\), \(X_{5}\) is on the extension line of \(X_{1}X_{4}\), and \(X_{5}\) is called the reflection point of \(X_{1}\) with respect to \(X_{4}\): \[X_{5}=X_{4}+\alpha(X_{4}-X_{1})=-\alpha X_{1}+(1+\alpha)X_{4}\] (9) where \(\alpha\) is the reflection coefficient, which equals 1 as usual. The geometric relationship is shown in Figure 2:
2. \(f(X_{5})\leq f(X_{3})\) denotes that that direction of searching is correct, the algorithm should keep going in this direction. Let \(\alpha=1.5\). If \(f(X_{6})\leq f(X_{5})\), \(X_{5}\) is replaced by \(X_{6}\) to form a new simplex, or \(X_{6}\) is dropped. Figure 2: The Simplex Method. 3. \(f(X_{3})\leq f(X_{5})\leq f(X_{2})\) means that the searching is going in the right direction, but doesn't need to expand. 4. \(f(X_{2})\leq f(X_{5})\leq f(X_{1})\) demonstrates that \(X_{5}\) has gone too far to need to be retracted. 5. If \(f(X_{5})>f(X_{1})\), \(X_{1},X_{2}\) need to be retracted toward \(X_{3}\). * Aggregate and Disperse Operator  
The purpose of the aggregate operation is to provide guidance for the population gathering in a potential direction during the iteration. With this operation, the algorithm can increase the convergence speed in the initial stage and strengthen the local search ability in the later stage. \(x_{global}\), \(x_{best}\), \(X_{k}\) are three given solutions, the worst individual is moved toward the others. In order to accelerate the convergence speed, the elitist solution \(x_{ebest}\) and the global optimal solution \(x_{global}\) are used to guide the search process. The parameter setting is \(\alpha\in(0,2)\)[31] to ensure that the new solution maintains convergence while moving toward a better direction generally. \(\alpha\) can be considered as the punishment parameter for the poor individual, and -\(\alpha\) is the encouragement parameter. \(\alpha\in(0,1)\) denotes that the better solution is not found, so the influence of the original strategy should be moderately weakened. \(\alpha\in(1,2)\) denotes that the original strategy should be strengthen. The parameters are transformed to form a convex combination to avoid negative weight in the later stage. \[X_{5}=(1-\beta)X_{1}+\beta(\varphi X_{2}+(1-\varphi)X_{3}) \tag{10}\]  
The simplex method and ABC are fused in the two-dimensional coordinate space in Figure 3:  
\[X_{i}=(1-\beta)X_{k}+\beta\Big{(}\varphi X_{elite}+(1-\varphi)X_{global}\Big{)} \tag{11}\]  
where \(\beta\in(0,\ 2),\ \varphi\in(0,\ 1)\). The vectors OA, OB, and OC represent the global optimal solution, elite solution, and ordinary individual respectively. If \(OE=\beta OA+(1-\beta)OB\), according to convex combination analysis, the point \(E\) lies on the line segment \(AB\), and \(OF\) ends up in the triangle \(\Delta A^{\prime}B^{\prime}C\).