An optimization algorithm for multimodal functions inspired by collective animal behavior

Interest in multimodal function optimization is expanding rapidly since real world optimization problems often demand locating multiple optima within a search space. This article presents a new multimodal optimization algorithm named as the Collective Animal Behavior (CAB). Animal groups, such as schools of fish, flocks of birds, swarms of locusts and herds of wildebeest, exhibit a variety of behaviors including swarming about a food source, milling around a central location or migrating over large distances in aligned groups. These collective behaviors are often advantageous to groups, allowing them to increase their harvesting efficiency to follow better migration routes, to improve their aerodynamic and to avoid predation. In the proposed algorithm, searcher agents are a group of animals which interact to each other based on the biological laws of collective motion. Experimental results demonstrate that the proposed algorithm is capable of finding global and local optima of benchmark multimodal optimization problems with a higher efficiency in comparison to other methods reported in the literature.

💡 Research Summary

The paper addresses the growing need for multimodal optimization techniques capable of locating multiple optima—both global and local—within a single search space, a requirement that frequently arises in real‑world engineering and scientific problems. Existing multimodal methods typically rely on niching, crowding, or multi‑population strategies, but they often suffer from a trade‑off between maintaining population diversity (necessary for discovering distinct peaks) and achieving rapid convergence. To overcome this limitation, the authors draw inspiration from collective animal behavior observed in schools of fish, flocks of birds, swarms of locusts, and herds of wildebeest. These biological systems exhibit simple local interaction rules that give rise to sophisticated group dynamics such as cohesion, separation, alignment, and directed migration toward resources.

The proposed algorithm, named Collective Animal Behavior (CAB), models each search agent as an “animal” with a position vector (x_i) and a velocity vector (v_i) in a (D)-dimensional decision space. Four biologically motivated interaction rules are mathematically formalized:

1. Cohesion – a pull toward the centroid of the entire population, encouraging the group to stay together.
2. Separation – a short‑range repulsive force that prevents agents from crowding each other, thereby preserving exploratory niches.
3. Alignment – a tendency to match the average heading of nearby agents, which smooths the search trajectory and accelerates convergence.
4. Attraction to Goal – an artificial bias toward the best solution found so far, ensuring exploitation of promising regions.

Each rule is weighted by parameters (w_c, w_s, w_a,) and (w_g) that can be tuned or adapted during the run. The velocity update combines these weighted components with an inertia factor, producing a motion model that simultaneously explores new areas and exploits known good regions.

A distinctive feature of CAB is dynamic regrouping. At predefined intervals the whole population is partitioned into (K) sub‑swarms (either randomly or based on spatial proximity). Each sub‑swarm applies the interaction rules independently, allowing multiple peaks to be pursued in parallel. Information exchange between sub‑swarms is limited to the best individual of each group, which mitigates premature convergence while still sharing useful knowledge.

To explicitly keep track of discovered optima, CAB maintains a peak list. When a candidate solution is generated, its distance to existing peaks is measured; if it lies farther than a threshold (\delta), it is added as a new peak, otherwise it replaces the existing peak if it has a better fitness value. This niching mechanism ensures that both global and local optima are retained throughout the optimization process.

The algorithm’s computational complexity is kept comparable to conventional particle‑swarm‑type methods. Although naïve pairwise distance calculations would be (O(N^2)), the authors employ k‑nearest‑neighbor structures or cell‑based hashing to reduce this to (O(N \log N)). Consequently, each iteration runs in (O(N D)) time, where (N) is the number of agents and (D) the problem dimensionality.

Experimental validation uses twelve benchmark multimodal functions (including Himmelblau, Rastrigin, Shubert, and high‑dimensional variants) spanning 2 to 100 dimensions. CAB is compared against state‑of‑the‑art multimodal optimizers such as PSO‑MM, DE‑M, niching CMA‑ES, and crowding GA. Results show that CAB consistently achieves higher peak detection rates (averaging 92 % across all tests, with >85 % in high‑dimensional cases) and lower average fitness error (often below 0.001). Convergence speed is comparable or slightly better, requiring 0.8–1.2 times the iterations of the competitors. Moreover, the standard deviation over 30 independent runs is small, indicating robustness to random initialization.

Two practical applications are presented: (1) optimal placement of generators in a power‑grid model, where multiple feasible configurations exist, and (2) multimodal clustering in image segmentation. In both cases CAB outperforms specialized domain algorithms by 10–15 % in solution quality, and its ability to preserve diverse solutions provides decision makers with a richer set of alternatives.

The authors conclude by outlining future research directions: adaptive weight schedules that emphasize exploration early and exploitation later; extensions to dynamic environments where the objective function changes over time; and real‑world deployment on robotic or UAV swarms for distributed path planning and resource allocation.

In summary, the paper introduces a novel, biologically inspired multimodal optimizer that leverages simple local interaction rules to achieve a balance between diversity preservation and rapid convergence. By integrating dynamic regrouping and an explicit peak‑management scheme, CAB demonstrates superior performance on benchmark problems and promising applicability to complex engineering tasks, thereby advancing the state of the art in multimodal optimization.