# Ellipsoidal K-Means: An Automatic Clustering Approach for Non-Uniform Data Distributions
## 3 Integrating Adaptive Ellipsoidal Clustering with Simulated Annealing Heuristic
### Simulated Annealing Heuristic  
Simulated annealing is an iterative search technique inspired by the physical process of annealing in solids. The process begins by heating a metal to a high temperature and then gradually cooling it, allowing the system to reach an approximate thermodynamic equilibrium at each stage. At this equilibrium, multiple configurations may exist, each corresponding to a specific energy level. The probability of accepting a transition from the current configuration to a new one depends on the energy difference between the two states. Simulated annealing was first introduced for optimization problems by [PERSON] et al. in [18]. Since its introduction, it has been widely applied to various combinatorial optimization problems, consistently delivering promising results across different instances.  
One of the primary advantages of the simulated annealing method is its ability to avoid local minima. Unlike traditional gradient-based methods, simulated annealing uses a stochastic search strategy that not only accepts changes that lead to a decrease in the objective function (such as the \(SF\)) but also occasionally accepts changes that increase it. This approach enhances the algorithm's ability to explore a broader solution space and avoid becoming stuck in local optima. Specifically, in our algorithm, increases in the objective function are accepted based on the probability given by \(\exp((f-f^{\prime})/T)\), where \(\Delta f=f(j)-f(i)\) represents the change in the objective function value, and \(T\) is the temperature parameter that governs the probability of accepting less favorable solutions.  
Algorithm 1 presents the fundamental steps of the simulated annealing (SA) procedure. It is structured as a foundational reference for the primary algorithm, Algorithm 2, to enhance clarity and coherence. By initially introducing the essential steps and underlying concepts, Algorithm 1 establishes a framework that facilitates understanding of SA's general approach. This layout also gives specific enhancements aimed at optimizing the method, thereby illustrating the rationale behind the design choices in Algorithm 2, which provides a comprehensive outline of the main SAELLC algorithm.  
```
Input :\(K\): number of centroids, \(c_{k}\): initial centers, \(T_{\text{max}}\): maximum temperature, \(T_{\text{min}}\): minimum temperature, \(\lambda\): cooling rate (\(0<\lambda<1\)). Output : Optimal cluster centers  Choose \(K\) centroids randomly as initial centers \(c_{k},k=1,\ldots,K\);  Set \(T=T_{\text{max}}\);  Set cooling schema \(\lambda\); while\(T>T_{min}\)do repeat  Select a neighbor solution of centers \(c_{k}+\Delta c_{k}\);  Calculate the new objective function value \(f\); if\(\Delta f=f^{\prime}-f<0\)then  Exchange old centers with new ones;  end if else if\(\exp\frac{-\Delta f}{T}>random[0,1]\)then  Exchange old with new solution;  end if untilmaximum iteration of epoch length;  Decrease \(T\) using annealing schedule \(T=\lambda\times T\);  end while
```  
**Algorithm 1**Pseudo-code for the simulated annealing clustering algorithm to allocate cluster centers.