Combinatorial optimization problems are encountered often in various real-world applications, including logistics, scheduling, and network design. These problems involve finding the best possible solution from a finite set of discrete options by maximizing or minimizing an objective function subject to specified constraints. In such problems, the number of feasible solutions increases exponentially with the problem size, making it nearly impossible to find optimal solutions. To tackle these problems, many heuristic and metaheuristic algorithms have been developed to efficiently obtain approximate solutions.
Recent study proposes a novel approach that improves stability and solution quality of chaotic search algorithms to solve optimization problems. Image credit: Tokyo University of Science
Chaotic search (CS) is among such algorithms that utilize chaotic dynamics to search for solutions. Chaotic dynamics follow precise rules but can appear unpredictable due to their extreme sensitivity to minuscule changes in initial parameters. Unlike purely stochastic methods, CS generates deterministic yet highly irregular search trajectories that can promote thorough exploration of the solution space. This approach can help the search process avoid becoming trapped in local solutions.
Despite its strong global exploration ability, the performance of CS algorithms is highly sensitive to several control parameters. When the parameters match a problem’s characteristics, CS works well, but even a slight mismatch can lead to unstable behavior. To improve robustness, researchers have previously extended the CS method with a parameter-tuning approach (CST), introducing heuristic feedback mechanisms. However, in CST, all parameters are uniformly updated according to global statistics, limiting adaptability and stability in complex problems.
To overcome these limitations, a research team led by Professor Tohru Ikeguchi from the Faculty of Engineering at Tokyo University of Science (TUS), Japan, proposed a new learning-based adaptive tuning method that integrates CS with particle swarm optimization (CSPSO). The team included third-year doctoral student Mr. Fengkai Guo from TUS, Associate Professor Takafumi Matsuura from Nippon Institute of Technology, and Professor Takayuki Kimura from Tokyo City University, Japan. Their study was published in Nonlinear Theory and Its Applications, IEICE (NOLTA), on July 01, 2026.
“In Particle Swarm Optimization (PSO), which draws inspiration from flocks of birds and ant colonies, a group of particles—referred to as a “swarm”—moves collectively through the search space, converging on promising regions while maintaining diversity,” explains Prof. Ikeguchi. “Owing to its relatively simple implementation and computational efficiency, PSO has been applied to many optimization problems. In our approach, PSO is utilized to dynamically control parameters of the chaotic neural network during searching, enhancing solution quality and robustness.”
In the proposed CSPSO approach, parameter tuning of CS is achieved externally using PSO. First, a swarm of particles, where each particle represents a candidate parameter vector, is initialized. For each particle, CS is performed, and the fitness of each particle is evaluated based on the obtained solution at the end of the run. Next, PSO updates each particle based on the fitness results. These steps are repeated until a specific condition is satisfied.
This iterative interaction essentially forms a two-layer optimization framework where the outer PSO layer efficiently and adaptively tunes parameters, thereby regulating the strength of the chaotic excitation, while the inner CS layer improves the solution using the parameters. By continuously adapting the parameters during the search process, the framework aims to maintain useful chaotic activity while promoting stable convergence.
The researchers tested the CSPSO method on capacitated vehicle routing problems (CVRP), a fundamental logistics optimization problem in which a fleet of vehicles must serve customers with known demands while respecting vehicle capacity limits. The results showed that CSPSO consistently achieved better solution quality and higher robustness compared with conventional CS and CST methods.
Notably, the algorithm remained stable over a wide range of PSO settings. Although CSPSO required more computational time than CST, the researchers point out that it is not easy to configure the parameters of chaotic neural networks in conventional CS and CST to achieve efficient search. Furthermore, given the enormous computational cost of exhaustively searching the parameter space, CSPSO offers a practical means of improving the performance of CS and CST.
Proposed chaotic search with particle search optimization: The proposed approach forms a two-layer optimization framework: the outer particle swarm optimization layer handles parameter tuning, while the inner chaotic search improves the solution using the tuned parameters. Image credit: Professor Tohru Ikeguchi from Tokyo University of Science, Japan/CC-BY-NC-ND 4.0
“In CSPSO, swarm-based learning absorbs the parameter tuning burden, reducing the need for careful manual calibration,” remarks Prof. Ikeguchi. “It provides an effective enhancement technique for CS, making it more flexible and adaptive to different scenarios, including shift scheduling, factory production planning and information technology networks.”
This approach could improve the efficiency and robustness of optimization methods used in applications such as logistics, transportation, and scheduling.
Source: Tokyo University of Science
