HN-CSA: Improved Cuckoo Search

• The paper presents HN-CSA as an improved variant that integrates multi-objective fitness, dynamic keyword weighting, and a hidden nest strategy to enhance exploration and exploitation.
• It reports significant performance gains in autonomous-cargo-ship retrieval, with improvements in Recall, Precision, F1-score, and Pareto-front hypervolume compared to standard CSA and NSGA-II.
• In Heston calibration, the adaptive discovery rate and strict bound projection enable rapid convergence with parameter errors below 10⁻⁴, ensuring effective, feasible solutions.

Improved Cuckoo Search Algorithm (HN-CSA) denotes a set of application-specific extensions of the canonical Cuckoo Search Algorithm (CSA) in which Lévy-flight search, nest replacement, and population management are modified to improve retrieval, calibration, or related optimization tasks. In the autonomous-cargo-ship literature, HN-CSA explicitly refers to a Hidden-Nest Cuckoo Search Algorithm that combines multi-objective fitness, dynamic keyword weighting, and temporary removal of the current best nest during search (Zhang et al., 18 Jul 2025). In Heston-model calibration for American options, an improved CSA is used with an adaptive discovery rate and strict bound projection for five-parameter nonlinear least-squares estimation (Haring et al., 2015).

1. Canonical algorithmic substrate

The standard CSA treats each nest as a candidate solution and evolves a population by Lévy flights plus replacement of a fraction of poor solutions. In the calibration formulation, the canonical update is written as

$x^{t+1} = x^t + \alpha\,\mathrm{Levy}(\lambda),$

where $\alpha>0$ is a step-size scalar and $\mathrm{Levy}(\lambda)$ draws steps from a Lévy distribution with exponent $\lambda\in(1,3)$. Each iteration generates new candidates, compares them against randomly chosen nests, abandons a fraction $p_a$ of the worst nests, and retains the current best solution (Haring et al., 2015).

A closely related formulation appears in multilevel image thresholding, where CSA is organized into a Lévy-flight exploitation step and a breeding step controlled by discovery probability $p_a$. In that setting, the exploitation move is written as

$$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$

followed by a randomized-walk breeding update

$$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$

with fitness-based replacement (Huang et al., 2020).

These formulations establish the common substrate from which improved variants depart: stochastic long-range search via Lévy flights, partial abandonment through $p_a$, and competition among nests. The principal differences among improved variants therefore lie in how fitness is defined, how neighborhood or elite information is injected, and how feasibility is enforced.

2. Hidden-Nest HN-CSA for literature retrieval

In the autonomous cargo ship setting, HN-CSA is introduced as part of a hybrid feature-fusion framework for constructing a graph-structured dataset of failure modes (Zhang et al., 18 Jul 2025). Relative to standard CSA, three modifications are specified.

The first modification is a multi-objective fitness definition. Standard deviation fitness $f_1$ enforces balance among keyword occurrences, and average correlation fitness $\alpha>0$0 rewards overall relevance of the keyword set to the corpus. The total fitness is the vector $\alpha>0$1, and nests are compared via Pareto dominance and hypervolume. The component formulas are

$\alpha>0$2

and

$\alpha>0$3

where $\alpha>0$4 is the weighted frequency of keyword $\alpha>0$5, $\alpha>0$6 is its mean across $\alpha>0$7, $\alpha>0$8 is the count of keyword $\alpha>0$9 in document $\mathrm{Levy}(\lambda)$0, and $\mathrm{Levy}(\lambda)$1 is the current keyword weight (Zhang et al., 18 Jul 2025).

The second modification is dynamic keyword weighting. At each iteration, the weight of keyword $\mathrm{Levy}(\lambda)$2 is updated by normalized frequency,

$\mathrm{Levy}(\lambda)$3

where $\mathrm{Levy}(\lambda)$4 is the raw frequency of keyword $\mathrm{Levy}(\lambda)$5 in the current retrieved set. This makes the search state-dependent: keyword importance is recomputed after each generation rather than fixed in advance (Zhang et al., 18 Jul 2025).

The third and defining modification is the hidden-nest strategy. After each generation, the single best nest $\mathrm{Levy}(\lambda)$6 is recorded and temporarily removed from the search in iteration $\mathrm{Levy}(\lambda)$7. The best nest is selected as

$\mathrm{Levy}(\lambda)$8

All other nests then update themselves by Lévy-flight or basic exploration/exploitation, and the hidden nest is reintroduced at the end of the next iteration. The stated purpose is to guarantee diversity and avoid premature convergence (Zhang et al., 18 Jul 2025).

The iteration-level workflow is therefore: initialize keyword subsets as nests, compute weights and fitness, identify the current best nest, hide it, update the active population, replace worst nests with new random subsets with probability $\mathrm{Levy}(\lambda)$9, split the active population into exploration and exploitation subsets, reinsert the hidden nest, update keyword weights, and recompute fitness. Hypervolume is used for Pareto-front comparison through

$\lambda\in(1,3)$0

where $\lambda\in(1,3)$1 is a reference point below all solutions (Zhang et al., 18 Jul 2025).

3. Adaptive-discovery improved CSA for Heston calibration

In American-option calibration, the improved CSA is used to fit the five-parameter Heston model. The parameter vector is

$\lambda\in(1,3)$2

where $\lambda\in(1,3)$3 is the initial volatility, $\lambda\in(1,3)$4 is the volatility of variance, $\lambda\in(1,3)$5 is the mean-reversion rate, $\lambda\in(1,3)$6 is the long-run variance, and $\lambda\in(1,3)$7 is the correlation between spot and variance. Calibration is posed as the nonlinear least-squares problem

$\lambda\in(1,3)$8

subject to box constraints on all five parameters (Haring et al., 2015).

The principal improvement in this formulation is an adaptive discovery rate. Instead of a fixed $\lambda\in(1,3)$9, the algorithm uses a linearly decreasing schedule,

$p_a$0

with $p_a$1 and $p_a$2. The intended effect is strong exploration early in the run and focused exploitation later in the run (Haring et al., 2015).

The second improvement is strict constraint handling by projection. After each Lévy-flight step,

$p_a$3

each component is clamped to its feasible interval:

$p_a$4

The exposition states explicitly that no local-search hybrid or modified Lévy distribution was used beyond the adaptive-$p_a$5 scheme and strict bound enforcement (Haring et al., 2015).

The calibration-specific pseudocode retains the canonical CSA structure—population initialization, Lévy-flight proposal, greedy selection, abandonment of worst nests, and memorization of the best solution—while replacing the fixed discovery rate with $p_a$6 and inserting projection after each update. This yields an improved CSA specialized for bounded continuous optimization rather than the Pareto-based discrete keyword-subset search used in the hidden-nest formulation.

4. Parameterization and computational regimes

The two improved CSA formulations use markedly different parameter settings because they target different search spaces and objective structures.

Aspect	Hidden-Nest retrieval (Zhang et al., 18 Jul 2025)	Heston calibration (Haring et al., 2015)
Population	$p_a$7 nests	$p_a$8 nests
Iteration budget	$p_a$9	$p_a$0 or until $p_a$1
Lévy setting	$p_a$2, $p_a$3 (fixed)	$p_a$4, $p_a$5
Discovery rate	$p_a$6	adaptive $p_a$7 from $p_a$8 to $p_a$9
Representation	keyword subset size 10–15	$$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$0
Additional control	typically 80% explore, 20% exploit	projection to bounds after each update

In the retrieval formulation, all methods were run for 100 generations, and the corpus ultimately comprised 415 candidate papers whose relevance to ACS component failure was judged by domain experts (Zhang et al., 18 Jul 2025). In the calibration formulation, each iteration evaluates up to $$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$1 candidate Lévy steps plus up to $$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$2 new random nests, giving $$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$3 model calls per iteration; because each objective evaluation requires pricing $$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$4 options, the per-iteration cost is $$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$5, and the stated total complexity is $$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$6 (Haring et al., 2015).

This suggests that the term “improved CSA” is not tied to a single parameter regime. In one case the algorithm is tuned for multi-objective discrete retrieval over keyword subsets; in the other it is tuned for bounded continuous calibration in $$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$7.

5. Empirical behavior and reported results

In the literature-retrieval experiment for autonomous cargo ship component failure, performance was measured by Recall, Precision, and F1-score. The reported results are: Keyword-only $$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$8, NSGA-II $$\mathbf{v} = \mathbf{x}_i + \alpha\,L(\lambda)\,(\mathbf{x}_i^{\rm best}-\mathbf{x}_i),$$9, CSA $$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$0, and HN-CSA $$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$1. The paper states that HN-CSA exceeds standard CSA by $$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$2 and exceeds NSGA-II by $$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$3, corresponding to the reported 3.4 percentage points and 7.1 percentage points. Pareto-front hypervolume after 100 iterations is reported as 0.145 for NSGA-II, 0.147 for CSA, and 0.153 for HN-CSA, which is described as a 5.5% improvement over NSGA-II and a 4.1% improvement over CSA in hypervolume coverage (Zhang et al., 18 Jul 2025).

The retrieval procedure is embedded in a larger graph-structured failure-analysis pipeline. The resulting dataset covers 12 systems, 1,262 failure modes, and 6,150 propagation paths. A hierarchical feature fusion framework uses Word2Vec encoding for subsystem/component features, BERT-KPCA for failure modes/reasons, and Sentence-BERT to quantify the semantic association between failure impact and emergency decision-making. Validation results report a GATE-GNN classification accuracy of 0.735, a silhouette coefficient of 0.641, and an F1 score of 0.93 for the Shore-based Meteorological Service System in label prediction (Zhang et al., 18 Jul 2025).

A common point of confusion is the location of the reported gains. In this autonomous-cargo-ship study, the 7.1% and 3.4% improvements refer to the literature-retrieval experiment, whereas the downstream graph-analysis metrics are reported separately. The abstract links HN-CSA to retrieval efficiency, while the GATE-GNN accuracy, silhouette coefficient, and subsystem F1 score characterize later stages of the feature-fusion and graph-learning pipeline (Zhang et al., 18 Jul 2025).

In Heston calibration, the improved CSA was tested on four artificial parameter sets using 84 prices per set. For all four cases, the recovered parameters were very close to the true values and the final objective was reported as approximately $$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$4. Convergence plots showed rapid descent of $$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$5 in the first 100–200 iterations, stability across nest sizes 20, 30, and 50, and a slight speed-accuracy trade-off when varying the number of input prices. The exposition also states that no direct head-to-head comparison with a fixed-$$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$6 CSA was carried out, even though the adaptive schedule was described as consistently delivering larger early-stage exploration and fine late-stage convergence with parameter error below $$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$7. The reported implementation in R with a compiled C++ pricer completed one full calibration—84 prices, 500 iterations, 20 nests—in under 15 seconds on a modern desktop (Haring et al., 2015).

6. Relation to other improved cuckoo-search variants

The broader arXiv literature on improved cuckoo-search methods includes variants that alter the information topology rather than the elite-management or constraint-handling scheme. A representative example is the Fully Informed Cuckoo Search (FICS) for multilevel image thresholding, which uses a ring topology so that each nest learns from a neighborhood of size $$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$8. In that method, neighbor fitnesses define weights

$$\mathbf{v} = \mathbf{x}_i + \delta(\mathbf{x}_{i_1}-\mathbf{x}_{i_2}), \qquad \delta\sim U(0,1),$$9

and the fully informed aggregate is

$p_a$0

The breeding step then becomes

$p_a$1

which the paper presents as a way to improve cooperation and learning efficiency in the population (Huang et al., 2020).

Empirically, FICS achieved the best mean value in 39 out of 40 cases, ranked first in Friedman tests across threshold levels $p_a$2, and was reported as the highest or statistically tied best method in 11 out of 16 PSNR/SSIM scenarios. Its parameterization also differs from both HN-CSA formulations: population size $p_a$3, $p_a$4, $p_a$5 for FICS, and a maximum function-evaluation budget of $p_a$6 (Huang et al., 2020).

This comparison clarifies an important terminological point. Across these sources, “improved Cuckoo Search Algorithm” does not denote a single universally fixed operator set. The hidden-nest method emphasizes Pareto-based elite removal and reinsertion; the Heston-calibration method emphasizes adaptive abandonment and projection; the fully informed method emphasizes neighborhood aggregation on a ring topology. A plausible implication is that HN-CSA is best understood as a family resemblance centered on modifying CSA’s exploration-exploitation balance, information flow, or feasibility management, with the exact mechanism determined by the application domain.