Ecological Cycle Optimizer (ECO)

• ECO is a nature-inspired metaheuristic for continuous optimization that mimics ecological energy and material cycling through producer, consumer, and decomposer roles.
• It employs a structured role division where producers form an elite archive, consumers perform guided motion via predation, and decomposers inject both local and global variation.
• Benchmark results show ECO's competitive performance on unconstrained and constrained problems, highlighting its scalability, robustness, and balanced exploration–exploitation dynamics.

Searching arXiv for the primary ECO paper and closely related acronym variants to ground the article in the current literature. Ecological Cycle Optimizer (ECO) is a nature-inspired metaheuristic algorithm for global optimization proposed in 2025 and motivated by energy flow and material cycling in ecosystems (Ma et al., 28 Aug 2025). In ECO, the optimization process is interpreted as an ecosystem evolving toward a stable, balanced state, with candidate solutions assigned ecological roles as producers, consumers, and decomposers. The acronym is not unique in the literature: in a 2025 paper on human-behavior metaheuristics, ECO denotes the Educational Competition Optimizer rather than an ecological algorithm (Zhao et al., 11 Sep 2025), while other works use ecological optimization to describe thermodynamic operating criteria (Levario-Medina et al., 2020) or eco-driving trajectory optimization under the name ECO+ (Hadjigeorgiou et al., 5 Jun 2025). In the strict algorithmic sense, however, Ecological Cycle Optimizer denotes the producer–consumer–decomposer metaheuristic introduced for continuous global optimization (Ma et al., 28 Aug 2025).

1. Ecological model and problem representation

ECO is formulated for continuous optimization with decision vector dimension $D$, objective function $f(\cdot)$, lower and upper bounds $\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$, population size $N_{\text{pop}}$, and iteration index $k = 1,\dots,k_{\max}$ (Ma et al., 28 Aug 2025). Each individual is a vector $\mathbf{X}_i \in \mathbb{R}^D$, initialized uniformly in the feasible box by

$$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$

The population is partitioned into four fixed ecological roles: $$N_{\text{Pro}} = 0.20N_{\text{pop}},\quad N_{\text{Her}} = 0.30N_{\text{pop}},\quad N_{\text{Car}} = 0.30N_{\text{pop}},\quad N_{\text{Omn}} = 0.20N_{\text{pop}}.$$ These correspond to producers, herbivores, carnivores, and omnivores. The ecological analogy is explicit. Producers form the base of the “food chain” and act as a high-quality pool. Consumers move through the search space by predation on lower trophic levels. Decomposers break individuals down into “inorganic matter,” which generates new candidate solutions and feeds the producer pool.

Five operational rules define the population ecology. First, trophic levels are ordered as Herbivore $<$ Carnivore $<$ Omnivore. Second, predation is constrained: herbivores prey only on producers, carnivores only on herbivores, and omnivores prey on producers, herbivores, and carnivores, but not on the same trophic level. Third, at the end of each iteration, all producers and consumers are decomposed. Fourth, energy is inversely related to fitness: lower fitness implies higher energy. Fifth, survival is elitist at the individual level: if an update improves fitness, the new position is retained; otherwise the old one is kept (Ma et al., 28 Aug 2025).

This structure gives ECO a clear role decomposition. Producers behave as an elite archive, consumers implement guided motion, and decomposers inject both local and global variation.

2. Consumer predation and producer absorption

The consumer dynamics are based on a generic predation model: $f(\cdot)$0 where $f(\cdot)$1 is the predation factor vector and $f(\cdot)$2 denotes element-wise multiplication (Ma et al., 28 Aug 2025).

For each dimension $f(\cdot)$3,

$f(\cdot)$4

Early in the run, $f(\cdot)$5 can be negative or larger than $f(\cdot)$6, so the motion may move away from prey or overshoot it; later, $f(\cdot)$7, producing moderate attraction toward prey. The paper interprets $f(\cdot)$8 or $f(\cdot)$9 as exploration, and $\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$0 as exploitation (Ma et al., 28 Aug 2025).

Each consumer class uses roulette wheel selection over eligible prey, with probabilities proportional to inverse fitness. Herbivores select three producers, carnivores select three herbivores, and omnivores select one producer, one herbivore, and two carnivores. This creates a multi-elite predation mechanism in which different trophic levels receive different search guidance. Because roulette selection is probabilistic rather than deterministic, better individuals are favored without forcing all movement toward a single elite.

Producer updating follows a nutrient-absorption rule. Let $\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$1 denote decomposition outputs. The producer candidates are formed by sorting the union of current producers and decomposers: $\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$2 and the new producer set is the best $\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$3 individuals from this pool (Ma et al., 28 Aug 2025). The paper states that this forms a stable “elite archive” from which consumers feed.

3. Decomposer operators and ecological cycling

Decomposition is the distinctive third component of ECO. After producer and consumer updates, every individual is decomposed into a new candidate $\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$4. For each individual, the algorithm chooses optimal decomposition with probability $\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$5, and otherwise chooses between local random decomposition and global random decomposition with equal probability $\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$6 each (Ma et al., 28 Aug 2025).

Optimal decomposition centers search near the current best individual $\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$7. The neighborhood center is

$\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$8

and the decomposer is generated by

$\mathbf{Lb}, \mathbf{Ub} \in \mathbb{R}^D$9

The paper characterizes this as strong exploitation around the current best (Ma et al., 28 Aug 2025).

Local random decomposition explores radially around the current individual, with radius equal to the distance to the current best. With random direction components

$N_{\text{pop}}$0

the update is

$N_{\text{pop}}$1

The search radius is large early and shrinks as the population converges (Ma et al., 28 Aug 2025).

Global random decomposition performs a domain-wide random walk. Its coefficient is

$N_{\text{pop}}$2

with $N_{\text{pop}}$3 and $N_{\text{pop}}$4 as $N_{\text{pop}}$5. The resulting step is mixed with the current position through a random weight (Ma et al., 28 Aug 2025). Early iterations therefore permit large random jumps, whereas later iterations retain only small perturbations.

Taken together, the producer–consumer–decomposer cycle is the algorithm’s central design principle. Consumers supply guided motion, decomposition supplies both elite-biased and random regeneration, and producers preserve the highest-quality nutrients from that cycle.

4. Exploration–exploitation behavior and computational properties

ECO’s exploration–exploitation balance is attributed to four mechanisms: the dynamic predation factor $N_{\text{pop}}$6, the trophic role structure, roulette wheel prey selection, and the three decomposition modes (Ma et al., 28 Aug 2025). Consumers and global random decomposition support broad exploration. Elite-guided predation and optimal decomposition support intensification near promising regions. The magnitude controls in $N_{\text{pop}}$7, local decomposition radius, and global random walk coefficient shift the search naturally from exploration to exploitation as iterations proceed.

The paper quantifies diversity using the Hussain model: $N_{\text{pop}}$8 Exploration and exploitation percentages are then defined from $N_{\text{pop}}$9 relative to its maximum over the run (Ma et al., 28 Aug 2025). On 23 classic functions, ECO exhibits high exploration early, followed by a rapid transition toward exploitation, with exploitation approaching $k = 1,\dots,k_{\max}$0 as $k = 1,\dots,k_{\max}$1 increases.

Boundary handling is simple: if any component leaves $k = 1,\dots,k_{\max}$2, it is reinitialized using the same uniform sampling rule as the initial population (Ma et al., 28 Aug 2025). The algorithm is also parameter-light. Aside from $k = 1,\dots,k_{\max}$3 and $k = 1,\dots,k_{\max}$4, the paper states that there are no algorithm-specific hyperparameters to tune.

The reported complexity is

$k = 1,\dots,k_{\max}$5

where $k = 1,\dots,k_{\max}$6 denotes the cost of evaluating the objective function, and the space complexity is approximately $k = 1,\dots,k_{\max}$7 (Ma et al., 28 Aug 2025). The paper further reports that ECO’s runtime on CEC-2020 is comparable to CSA and WSO and significantly faster than INFO.

5. Benchmark results on unconstrained and constrained suites

The empirical evaluation spans classical unconstrained functions, the IEEE CEC-2006 constrained set, large comparative pools on CEC-2014 and CEC-2017, and focused tests on CEC-2020 (Ma et al., 28 Aug 2025).

Benchmark setting	ECO result	Comparative note
23 classical unconstrained functions	Friedman mean rank 1.00	Best average fitness on all 23 functions
CEC-2006 constrained problems	Friedman mean rank 1.38	ECO and CS are the two strongest performers
CEC-2014 + CEC-2017, 51 algorithms	Combined mean rank 2.82	Ranked first overall
CEC-2020, $k = 1,\dots,k_{\max}$8	Rank 1 at all dimensions	Advantage increases with dimension

On the 23 classical functions, the setup uses $k = 1,\dots,k_{\max}$9, $\mathbf{X}_i \in \mathbb{R}^D$0, and 25 independent runs. ECO achieves the best average fitness on all 23 functions, with Friedman mean rank $\mathbf{X}_i \in \mathbb{R}^D$1. The Wilcoxon rank-sum comparisons report 19 wins, 4 ties, and 0 losses against CS; 22/1/0 against HS; 21/2/0 against PSO; 18/5/0 against GWO; and 18/5/0 against WOA (Ma et al., 28 Aug 2025).

On the 24 CEC-2006 constrained problems, with $\mathbf{X}_i \in \mathbb{R}^D$2 and 25 runs, ECO obtains Friedman mean rank $\mathbf{X}_i \in \mathbb{R}^D$3. The paper states that ECO and CS are the two strongest performers, often reaching identical best values. Against CS, the Wilcoxon counts are 9 wins, 7 ties, and 8 losses, while ECO dominates HS, GWO, and WOA more clearly (Ma et al., 28 Aug 2025).

The broadest comparison uses 51 algorithms on CEC-2014 and CEC-2017 at $\mathbf{X}_i \in \mathbb{R}^D$4, $\mathbf{X}_i \in \mathbb{R}^D$5, and 51 runs. ECO attains the best combined Friedman mean rank, $\mathbf{X}_i \in \mathbb{R}^D$6, ahead of ARO, CFOA, CSA, INFO, and WSO (Ma et al., 28 Aug 2025). The focused CEC-2020 study then compares ECO against ARO, CFOA, CSA, WSO, and INFO at $\mathbf{X}_i \in \mathbb{R}^D$7. ECO ranks first at all four dimensions, with rank values $\mathbf{X}_i \in \mathbb{R}^D$8, $\mathbf{X}_i \in \mathbb{R}^D$9, $$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$0, and $$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$1, respectively. The paper emphasizes that ECO’s advantage increases with dimension, which it interprets as strong scalability and robustness on complex landscapes (Ma et al., 28 Aug 2025).

6. Engineering applications, strengths, and limitations

ECO is also tested on five constrained engineering problems from the CEC-2020-RW test suite: RC15 speed reducer weight minimization, RC17 tension/compression spring design, RC19 welded beam design, RC20 three-bar truss design, and RC31 gear train design (Ma et al., 28 Aug 2025). Across these five problems, Friedman mean ranks are reported as $$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$2 for COLSHADE, $$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$3 for EnMODE, $$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$4 for SASS, and $$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$5 for ECO, placing ECO among the best-performing methods in that comparison.

Several individual results are exact or essentially exact. For RC15, ECO reaches the same best value,

$$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$6

as SASS, EnMODE, NSM-SFS, FDB-SFS, FDB-AGDE, esCMAgES, L-SHADE, COLSHADE, and LRFDB-COA. For RC19, all algorithms including ECO achieve

$$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$7

For RC20, all algorithms reach

$$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$8

and for RC31 ECO attains

$$\mathbf{X}_i = \mathbf{Lb} + \text{rand} \cdot (\mathbf{Ub} - \mathbf{Lb}), \quad i = 1,\dots,N_{\text{pop}}.$$9

which the paper describes as essentially perfect matching (Ma et al., 28 Aug 2025).

The paper also states an important caveat about constraints. ECO does not introduce a special constraint-handling mechanism; the experiments follow the CEC technical reports, and infeasible solutions are discarded or treated as infeasible, with ECO’s survival rule applied assuming feasibility (Ma et al., 28 Aug 2025). This is consistent with the broader limitations explicitly listed in the paper: ECO as presented is focused on continuous single-objective problems, the effect of the fixed $$N_{\text{Pro}} = 0.20N_{\text{pop}},\quad N_{\text{Her}} = 0.30N_{\text{pop}},\quad N_{\text{Car}} = 0.30N_{\text{pop}},\quad N_{\text{Omn}} = 0.20N_{\text{pop}}.$$0 role split is not systematically studied, the design is random-heavy, and no formal convergence proof is given.

At the same time, the paper identifies several strengths: high performance across diverse suites, automatic exploration–exploitation management, parameter simplicity, robustness to local optima, scalability, and conceptual clarity in the mapping between ecological roles and search operators (Ma et al., 28 Aug 2025). The only future direction stated explicitly is multi-objective ECO.

7. Nomenclature and related ECO usages

The term “ECO” is overloaded, and this has direct implications for literature searches. In the 2025 paper “An improved educational competition optimizer with multi-covariance learning operators for global optimization problems,” ECO refers to the Educational Competition Optimizer, a human-behavior metaheuristic based on competition in education systems rather than ecological cycles (Zhao et al., 11 Sep 2025). That work retains the acronym ECO but is conceptually unrelated to the producer–consumer–decomposer model.

A second usage appears in finite-time thermodynamics. “Energetic optimization considering a generalization of the ecological criterion in traditional simple-cycle and combined cycle power plants” studies a generalized ecological objective

$$N_{\text{Pro}} = 0.20N_{\text{pop}},\quad N_{\text{Her}} = 0.30N_{\text{pop}},\quad N_{\text{Car}} = 0.30N_{\text{pop}},\quad N_{\text{Omn}} = 0.20N_{\text{pop}}.$$1

with $$N_{\text{Pro}} = 0.20N_{\text{pop}},\quad N_{\text{Her}} = 0.30N_{\text{pop}},\quad N_{\text{Car}} = 0.30N_{\text{pop}},\quad N_{\text{Omn}} = 0.20N_{\text{pop}}.$$2, to identify operating regimes between maximum power and maximum efficiency in irreversible Curzon–Ahlborn-type models (Levario-Medina et al., 2020). This is an ecological optimization framework for thermodynamic cycles, not a population-based metaheuristic.

A third usage appears in autonomous driving. “Energy Consumption Optimization for Autonomous Vehicles via Positive Control Input Minimization” introduces ECO+, a convex, time-based trajectory optimization framework that minimizes Positive Control Input rather than using an evolutionary search scheme (Hadjigeorgiou et al., 5 Jun 2025). The paper explicitly notes that ECO+ is not a metaheuristic or evolutionary algorithm.

Within the broader ecosystem-inspired optimization literature, an earlier conceptual precursor is the ecosystem-oriented distributed evolutionary computing framework based on habitats, gene-pools, migration probabilities, and local populations that evolve candidate gene-sets in response to requests (Briscoe et al., 2012). That framework can be viewed as a direct basis for an ecological optimizer in the distributed sense, but it is architecturally distinct from the 2025 Ecological Cycle Optimizer.

In current usage, therefore, “Ecological Cycle Optimizer” most precisely denotes the 2025 producer–consumer–decomposer metaheuristic for continuous global optimization (Ma et al., 28 Aug 2025), while similarly named ECO frameworks in education, thermodynamics, and eco-driving refer to different optimization paradigms.