Enhanced Particle Swarm Optimization

• Improved Particle Swarm Optimization algorithms are modified PSO frameworks that enhance solution quality, convergence speed, and computational efficiency through tailored update mechanisms.
• They incorporate adaptive parameter tuning, heterogeneous dynamics, hybrid metaheuristics, and surrogate modeling to effectively address high-dimensional, multimodal, and real-time optimization challenges.
• Empirical benchmarks demonstrate significant reductions in computational complexity—up to 57% less per-iteration operations—while maintaining or improving overall optimization performance.

An improved Particle Swarm Optimization (PSO) algorithm denotes any modification to the canonical PSO framework that enhances a particular facet of optimization—such as solution quality, convergence rate, global search capability, parameter adaptation, or computational complexity—through systematic algorithmic, mathematical, or hybridization strategies. Improved PSO algorithms have introduced paradigm shifts in swarm initialization, velocity and position update rules, parameter management, adaptation mechanisms, hybrid metaheuristics, surrogate modeling, heterogeneous population strategies, learning-from-history enabled behaviors, as well as domain-specific enhancements for real-time computation, large-scale combinatorial settings, and black-box, multimodal optimization.

1. Strategies for Performance and Complexity Improvement

Several improved PSO variants have targeted the trade-off between optimization performance and computational complexity. For instance, the Accelerated CLPSO (ACLPSO) algorithm modifies the comprehensive learning PSO (CLPSO) by implementing an event-triggered update mechanism: dimensions for which the difference $|\hat{p}_k^d - x_k^d(i)|$ is less than a given threshold $\gamma$ are not updated, i.e., $c_k^d(i) = 0$ in the velocity update equation. Only significant errors trigger a costly tensor update, thereby reducing the number of per-iteration multiplications to 43–67% of standard CLPSO while incurring only a modest performance loss—enabling application to real-time signal processing and tracking contexts (Saeed et al., 2013).

Approach	Key Modification	Complexity Impact
ACLPSO	Event-triggered dimensional updates	-33% to -57% ops
CLPSO	Full-dimensional comprehensive learning	High cost

Such frameworks exemplify the principle of tailoring update frequency and locality to problem structure to control resource requirements without unacceptable accuracy degradation.

2. Heterogeneous and Adaptive Swarm Behaviors

Adopting heterogeneous learning and adaptive parameterization has yielded substantial gains in solution robustness on multimodal and high-dimensional problems. The Heterogeneous Strategy PSO (HSPSO) divides particles between singly informed (SI) and fully informed (FI) dynamics, governed by a proportion parameter $\lambda$. SI particles rely on neighborhood-best information and maintain diversity, while FI particles use the best experience of all their neighbors, enhancing convergence. Experimental tuning of $\lambda$ demonstrates superior solutions to canonical (SIPSO) and fully informed (FIPSO) baselines across both unimodal and multimodal benchmarks; this is attributed to an optimized exploration–exploitation balance (Du et al., 2016).

Particle Type	Information Used	Role
SI	Own and neighbor-best	Exploration
FI	All neighbor-best	Exploitation

Adaptation can also extend to online adjustment of control coefficients using fuzzy inference or reinforcement learning. Reinforcement learning-based parameter adaptation (RLAM), for example, leverages actor-critic frameworks to adjust inertia and acceleration weights based on swarm diversity, iteration progress, and historical improvement, as quantified by sin-encoded state vectors (ShiYuan, 2022). This results in consistent ranking improvements across a large suite of CEC2013 benchmark functions.

3. Surrogate-Model Assisted Optimization

Directed PSO (DPSO) with Gaussian-process-based function forecasting integrates a probabilistic surrogate model into the PSO update, augmenting the velocity with a term targeting the minimizer of the surrogate’s posterior mean or its high-uncertainty regions. Particles are thus guided not only by sampled optima but also by forecasted promising domains, resulting in faster convergence and superior exploitation/exploration dynamics relative to canonical PSO or evolutionary surrogate-assisted baselines, with statistical significance validated on CEC2013 test functions (Jakubik et al., 2021).

4. Hybridization and Local Search Integration

Algorithmic hybridization, wherein PSO is combined with metaheuristics such as Differential Evolution (DE), polynomial or quadratic model-based local search, or genetic algorithm-based components, offers increased capacity to escape local minima and leverage different search paradigms for different optimization phases. The M-GAPSO variant exemplifies this, dynamically switching between PSO, DE, or locally fitted model-based steps, guided by a moving-average measure of improvement. Archived samples are stored with R-Tree structures for quick reuse and local modeling—improving competitive performance metrics in BBOB and BBComp benchmarks, particularly in low- to moderate-dimensional problems (Okulewicz et al., 2020).

5. Discrete, Combinatorial, and Domain-Specific Extensions

Improved discrete PSO variants such as OMPCDPSO introduce multi-parent crossover and local exploitation via onlooker bees (Bee Algorithm operators), enabling high efficacy in allocation and NP-hard combinatorial problems (Zibaei et al., 15 Mar 2024). For real-time multi-agent trajectory planning in dynamic UAV swarms, PE-PSO introduces persistent exploration through periodic partial reinitialization and entropy-based adaptive parameter adjustment, ensuring diversity and responsiveness. This is further embedded in a multi-agent framework with GA-based task allocation, B-spline trajectory parameterization, parallelized PE-PSO planners, and decentralized control for scalable, low-latency swarm operations (Li et al., 18 Jul 2025).

6. Theoretical Reformulation and Trajectory Analysis

PSO algorithms have also been recast in unified frameworks to facilitate systematic analysis and custom variant design. Reformulated PSO (RePSO) expresses the position update as a second-order difference equation,

$$x^{(t+1)} = x^{(t)} + \omega(x^{(t)} - x^{(t-1)}) + \varphi(p - x^{(t)}),$$

where closed-form solutions allow classification and control of monotonic, oscillatory, and zigzagging behaviors, and reveal precise conditions for convergence ($\omega < 1$ and $\omega > (\varphi/2)-1$). Particle-level attribute assignment for inertia, acceleration, sociometry (topology), and constraint-handling permits construction of highly heterogeneous and flexible optimization systems (Innocente, 2021). This theoretical generality has underpinned the design of new high-performance, application-adaptive PSO variants.

7. Benchmark Validation and Application Domains

Empirical evaluation across a wide array of domains—ranging from digital filter design, optimal scheduling, real-time swarm robotics, high-dimensional function optimization, to medical diagnostics (Transformer hyperparameter tuning for heart disease prediction)—demonstrates that improved PSO algorithms consistently outperform canonical versions—and, in many setups, contemporary metaheuristics (e.g., DE, GA, ABC, surrogate-assisted EA)—on metrics including convergence rate, final fitness, computation time, and solution reliability (Yi et al., 3 Dec 2024). The incorporation of advanced initialization (orthogonal arrays (Bala et al., 21 May 2024)), adaptive parameter tuning, and hybridization forms the basis for robust, scalable solvers that can be calibrated for real-time, high-dimensional, and non-convex problem settings.

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In summary, improved particle swarm optimization algorithms represent a diverse and rapidly evolving field focused on algorithmic modifications that ensure more rapid convergence, enhanced global search ability, greater solution stability, and scalable efficiency. These advances leverage a wide range of mathematical, algorithmic, and hybridization innovations, validated through competitive benchmarking and domain-specific deployments, and are underpinned by a growing theoretical foundation for systematic analysis and design.