# BiModalClust: Fused Data and Neighborhood Variation for Advanced K-Means Big Data Clustering
## 3 Variable Neighborhood Search  
### Main Concepts  
In the fields of computer science, artificial intelligence, and mathematical optimization, heuristics are indispensable tools for expediting problem-solving processes. They are particularly valuable when traditional methods are computationally prohibitive or when seeking exact solutions is infeasible due to the problem's inherent complexity. It is important to note, however, that heuristics do not guarantee the discovery of the optimal solution, categorizing them as approximate algorithms. These algorithms are adept at rapidly and efficiently generating solutions that closely approximate the optimal one. In some instances, they may even achieve the exact optimal solution, but they remain classified as heuristics until their output is formally proven to be optimal [18]. Expanding on this concept, metaheuristics provide flexible and adaptive frameworks for designing heuristics, enabling them to address a wide variety of combinatorial and global optimization problems [19]. Metaheuristics generalize the heuristic approach, introducing higher-level strategies to explore the solution space more effectively and avoid pitfalls such as premature convergence to suboptimal solutions. An optimization problem can be formally described as follows:\[\min f(x)\mid x\in\mathcal{F}\subseteq\mathcal{X}, \tag{2}\]  
where  
* \(\mathcal{X}\) represents the ambient solution space encompassing all potential solutions,
* \(\mathcal{F}\) denotes the feasible subset of \(\mathcal{X}\) where solutions satisfy predefined constraints,
* \(x\) is an individual feasible solution within \(\mathcal{F}\), and
* \(f\) is the real-valued objective function to be minimized. In the context of MSSC (1), the feasible solution space simplifies to \(\mathcal{F}=\mathcal{X}=\mathbb{R}^{n}\), reflecting the continuous multidimensional nature of the clustering problem. Variable Neighborhood Search (VNS), introduced by [PERSON] in 1997 [4], is a modern metaheuristic framework that provides a flexible and robust approach to solving combinatorial and continuous nonlinear global optimization problems. The strength of VNS lies in its systematic exploitation of neighborhood changes, enabling both descent to local minima and escape from the valleys containing them [4; 19; 20]. This method strategically explores distant neighborhoods of the current solution and moves to a new solution only when such movement yields an improvement in the objective function. VNS serves as a powerful tool for constructing innovative global search heuristics by integrating existing local search methods. Its effectiveness is grounded in the following foundational principles:  
* A solution that is a local optimum under one neighborhood structure may not remain optimal when assessed using a different neighborhood. * A globally optimal solution is also a local optimum for all possible neighborhood structures. * For many optimization problems, local optima across one or multiple neighborhoods tend to be located in close proximity to one another. Fact 1 establishes the basis for using diverse and complex moves to uncover local optima across multiple neighborhood structures. Fact 2 suggests that incorporating a larger variety of neighborhoods into the search process increases the likelihood of discovering a global optimum, particularly when the current local optima are suboptimal. Together, these insights propose an intriguing strategy: a solution that is a local optimum across several neighborhood structures has a higher probability of being globally optimal compared to one that is optimal within a single neighborhood [21]. Fact 3, which arises primarily from empirical observations, underscores that local optima often provide valuable insights into the characteristics of the global optimum. In essence, local optima in optimization problems tend to share structural similarities. This phenomenon highlights the utility of thoroughly exploring the neighborhoods of a given local optimum.