Optimization has become an integral part of modern engineering design, playing an increasingly crucial role in practical applications. Real-world optimization problems are often non-convex and nonlinear, involving multiple local optima. In many cases, the design variables span several orders of magnitude, and smoothness assumptions such as differentiability of objective and constraint functions may not hold. Furthermore, as design requirements grow more stringent and engineered systems become larger and more complex, large-scale optimization problems (LSOPs), in which the number of design variables can reach into the hundreds, are now commonly encountered [1,2]. These trends have led to a growing demand for optimization algorithms that can efficiently address such large and challenging problem settings.

Classical mathematical programming approaches have attempted to address such challenges through problem reformulation [3], or by leveraging search strategies to escape from local optima [4]. However, for problems involving discontinuous, non-differentiable, or highly non-convex objective or constraint functions, gradient-based deterministic methods often fail to provide satisfactory solutions.

In this context, evolutionary computation (EC) has attracted considerable attention as a more general and flexible optimization paradigm [5,6]. EC refers to a family of population-based metaheuristic algorithms that simulate biological evolution and collective behaviors, treating candidate solutions as individuals that evolve through genetic variation and cooperative interactions. Representative approaches include Particle Swarm Optimization (PSO), inspired by swarm intelligence (SI) [7,8], and the Genetic Algorithm (GA), introduced by Holland [9]. These methods do not rely on gradient information, making them particularly well-suited to handling nonlinear, discontinuous, and non-convex problems. Furthermore, their ability to explore multiple candidate solutions simultaneously increases the likelihood of locating globally optimal solutions. As a result, EC-based algorithms have demonstrated considerable success across a wide range of application domains [10-12].

The effectiveness of EC largely depends on two complementary abilities: exploration and exploitation. Exploration refers to the algorithm’s capacity to broadly investigate the search space and reach diverse regions, while exploitation denotes its ability to refine solutions within promising areas. Achieving a well-balanced trade-off between these two components is critical for both convergence efficiency and solution quality [13,14]. Consequently, many EC variants incorporate strategies explicitly designed to manage this balance [15,16]. Despite these advances, many EC-based methods still face a limitation: without specific enhancements, they tend to be effective only for problems with a relatively small number of design variables. As the dimensionality increases, their search efficiency deteriorates significantly [17,18]. A primary contributing factor is premature convergence, where the population rapidly collapses around a local optima, leading to a loss of diversity and stagnation. This issue is particularly pronounced in high-dimensional search spaces, where the number of local optima increases exponentially, and structural exploration becomes more challenging. To address the limitations, various EC variants have been proposed. Among PSO-based approaches, two examples are CRI-PSO [19] (PSO with rotational invariance using correlativity) and FDPSO [20] (Forked Divergence PSO). CRI-PSO improves global searchability by reconstructing the search space through rotationally invariant coordinate transformations derived from particle (search point) trajectories. FDPSO employs dimensionality reduction, allowing each particle to iteratively optimize only a selected subset of decision variables, thereby improving search efficiency in high-dimensional settings. Although both methods have demonstrated effectiveness for nonlinear continuous LSOPs, their designs primarily focus on subspace decomposition and lack an explicit mechanism to coordinate exploration and exploitation - an essential component of EC’s search behavior. Introducing a framework that explicitly incorporates this balance is therefore expected to further improve optimization performance.

Therefore, in this study, we propose a PSO-based algorithm that explicitly emphasizes the balance between exploration and exploitation in the context of nonlinear continuous LSOPs. The proposed method integrates two key concepts:

• Dimensionality reduction: Building on the FDPSO framework, each particle dynamically selects and updates a subset of decision variables, enabling focused search in lower-dimensional subspaces to mitigate the curse of dimensionality.
• Role-based swarm division: The swarm is partitioned into two subpopulation groups with complementary roles, an exploration group dedicated to global search and an exploitation group responsible for local refinement. Each group employs distinct strategies to independently pursue its search objective, thereby achieving a parallel and coordinated balance between exploration and exploitation.

Based on the above design principles, the method proposed in this study is termed Forked Divergence Particle Swarm Optimization with Exploration and Exploitation (FDPSO-EE). The FDPSO-EE inherits the dimensionality reduction framework of the original FDPSO while simultaneously achieving both exploration and exploitation. The aim is to significantly enhance the search performance of PSO in large-scale optimization problems. This paper investigates the effectiveness of the proposed method through its application to three benchmark functions characterized by high dimensionality and complex landscapes. The characteristics of the FDPSO-EE are analyzed in detail, and its performance is evaluated in comparison with other existing algorithms.

For the full article, Please go through this link: https://www.pubtexto.com/pdf/?development-of-forked-divergence-particle-swarm-optimization-with-exploration-and-exploitation