Multi-Strategy Improved Snake Optimizer

• The paper [2507.14043] enhances SO by using adaptive random disturbance, Lévy flight, and elite leadership with Brownian motion to boost early exploration and improve UAV path planning.
• The paper [2507.15832] integrates low-discrepancy initialization, adaptive thresholds, dual mutation, and composite flight to optimize CNN-LSTM-Attention-AdaBoost for accurate trajectory prediction.
• Empirical results across benchmark tests and real-world applications show that MISO reduces errors, accelerates convergence, and maintains population diversity better than the baseline SO.

Searching arXiv for the specified MISO papers and closely related Snake Optimizer work. arXiv search query: "Multi-Strategy Improved Snake Optimizer" Multi-strategy Improved Snake Optimizer (MISO) denotes a class of metaheuristic algorithms that modify the Snake Optimizer (SO) by combining multiple mechanisms intended to rebalance exploration and exploitation, improve convergence speed, and reduce entrapment in local optima. In the 2025 arXiv literature, the acronym is used in at least two distinct but related senses: one paper formulates MISO through adaptive random disturbance, adaptive Lévy flight, and elite leadership with Brownian motion for UAV path planning and engineering design (Li et al., 18 Jul 2025), while another integrates good-point-set initialization, adaptive thresholds, dual mutation, and a composite flight function for hyperparameter optimization in a CNN-LSTM-Attention-AdaBoost trajectory-prediction system (Li, 21 Jul 2025). This suggests that MISO is best understood as a multi-strategy enhancement framework built on SO rather than a single invariant algorithmic specification.

1. Baseline: the Snake Optimizer

The Snake Optimizer mimics snakes’ foraging, mating, and fighting behaviors. In the formulation summarized in the UAV-path-planning paper, SO divides the search into exploration, corresponding to seeking food, and exploitation, corresponding to fighting or mating (Li et al., 18 Jul 2025). The method uses two global parameters: food quantity,

$$Q = C_1 \exp\!\bigl(\tfrac{t}{T}\bigr),\quad C_1=0.5,$$

and temperature,

$$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$

where $t$ is the current iteration and $T$ the maximum. Initialization is uniform in the search box,

$$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$

and the population is split evenly into male and female snakes (Li et al., 18 Jul 2025).

In exploration, when $Q<0.25$, each snake jumps to a random mate’s position plus a small perturbation:

$$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$

with $C_2=0.05$ and

$$A_m=\exp\!\bigl(f_{\mathrm{rand},m}-f_{i,m}\bigr),\quad A_f=\exp\!\bigl(f_{\mathrm{rand},f}-f_{i,f}\bigr).$$

In exploitation, when $Q\ge0.25$, hot conditions with $$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$0 move snakes toward the best food,

$$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$1

whereas colder conditions switch to fighting or mating updates (Li et al., 18 Jul 2025).

The trajectory-prediction paper presents the same baseline SO at a more general level, with population size $$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$2, dimension $$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$3, search-space bounds $$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$4, initialization

$$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$5

and fixed thresholds $$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$6 in the basic SO (Li, 21 Jul 2025). Across both accounts, SO is a population-based optimizer whose state transitions are controlled by food quantity, temperature, and sex-specific interaction rules.

2. Two 2025 MISO formulations

The 2025 arXiv record documents two different MISO constructions. The first, in "A multi-strategy improved snake optimizer for three-dimensional UAV path planning and engineering problems" (Li et al., 18 Jul 2025), introduces three enhancements to SO: Adaptive Random Disturbance (DSO), Adaptive Lévy Flight (LSO), and Elite Leadership + Brownian Motion (BSO). The second, in "Multi-Strategy Improved Snake Optimizer Accelerated CNN-LSTM-Attention-Adaboost for Trajectory Prediction" (Li, 21 Jul 2025), integrates four strategies: good-point-set initialization, adaptive threshold parameters, dual mutation, and a composite flight function.

Paper	MISO components	Main application
(Li et al., 18 Jul 2025)	DSO, LSO, BSO	UAV 3D path planning; 6 engineering design problems
(Li, 21 Jul 2025)	Good-point-set initialization, adaptive thresholds, dual mutation, composite flight	CNN-LSTM-Attention-AdaBoost hyperparameter tuning for 4D trajectory prediction

This divergence is methodologically important. Both papers retain SO as the base behavioral skeleton, but they modify different parts of the update loop and emphasize different failure modes. In (Li et al., 18 Jul 2025), the principal concerns are slow convergence speed and susceptibility to local optima. In (Li, 21 Jul 2025), the stated goal is to balance exploration and exploitation while improving optimizer performance on large-scale high-dimensional trajectory data. A plausible implication is that MISO should be treated as a design pattern for SO enhancement rather than a uniquely standardized optimizer.

3. Strategy set in the UAV-path-planning MISO

The UAV-path-planning formulation introduces an adaptive random disturbance strategy based on a sine function to alleviate the risk of getting trapped in a local optimum (Li et al., 18 Jul 2025). The disturbance factor is

$$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$7

This factor replaces the update multiplier in the SO transitions, including exploration and hot-exploitation updates. Because DF decreases as $$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$8, the paper characterizes it as boosting early exploration.

The same formulation adds adaptive Lévy flight by granting male leaders long-jump flights:

$$\mathit{Temp} = \exp\!\bigl(\tfrac{T - t}{T}\bigr),$$9

with

$t$0

The classical Lévy-distribution term is given with $t$1 and $t$2 (Li et al., 18 Jul 2025). In parallel, female leaders perform weighted Brownian walks to refine local search:

$t$3

where

$t$4

The corresponding pseudocode uses a two-stage schedule. For $t$5, it computes DF and applies disturbance-scaled SO transitions; for $t$6, it computes $t$7, $t$8, and $t$9, then updates males by the Lévy-flight equation and females by the Brownian equation (Li et al., 18 Jul 2025). Exploration versus exploitation is monitored via population diversity $T$0 and a split at $T$1. The paper also identifies a limitation: the extra parameters DF and CF require calibration, and the two-stage switching may be problem-dependent (Li et al., 18 Jul 2025).

4. Strategy set in the trajectory-prediction MISO

The trajectory-prediction formulation defines four key strategies intended to balance exploration and exploitation (Li, 21 Jul 2025). First, good-point-set initialization replaces uniform random initialization with a low-discrepancy set in $T$2:

$T$3

Its stated motivation is to ensure uniform coverage of the search space at $T$4.

Second, adaptive threshold parameters periodically adjust attraction and repulsion:

$T$5

with cycle $T$6. The stated motivation is to avoid premature convergence.

Third, the method introduces a dual mutation strategy. Main mutation uses Cauchy perturbation in early iterations,

$T$7

and Gaussian perturbation in late iterations,

$T$8

Auxiliary mutation comprises head chaos via a Logistic map,

$T$9

body fusion through midpoint crossover,

$$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$0

and tail splice,

$$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$1

Fourth, the composite flight function alternates between long-range jumps and local walks. Early iterations use adaptive Lévy flight,

$$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$2

whereas late iterations use a random walk,

$$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$3

with a switch at $$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$4 (Li, 21 Jul 2025). The integration order is explicit: after the standard SO exploration or development update, the algorithm applies adaptive parameters, composite flight, dual mutation on selected individuals, and boundary control. Recommended experimental settings are $$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$5 and $$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$6 for CEC2022 benchmark functions, $$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$7 and $$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$8 for CNN-LSTM-Attention-AdaBoost hyperparameter tuning, dual mutation rates $$X_i = X_{\min} + \mathrm{rand}\times (X_{\max}-X_{\min}),\quad i=1\ldots N,$$9 and $Q<0.25$0, Lévy exponent $Q<0.25$1, and random-walk scale $Q<0.25$2 (Li, 21 Jul 2025).

5. Computational properties and optimization behavior

Both 2025 formulations report the same asymptotic time complexity order as the base SO. The trajectory-prediction paper states a per-iteration cost of $Q<0.25$3 for updates plus $Q<0.25$4 fitness calls, giving total complexity $Q<0.25$5 (Li, 21 Jul 2025). The UAV-path-planning paper states that time complexity remains $Q<0.25$6, the same as SO, and adds that empirical CPU times are slightly lower than SO (Li et al., 18 Jul 2025).

The two papers also provide complementary theoretical interpretations of why the improvements matter. The trajectory-prediction paper concludes that the combination of low-discrepancy initialization, periodic adaptive thresholds, dual mutations, and composite flights allows MISO to maintain population diversity in early iterations and refine solutions smoothly in later iterations; it further states that MISO avoids stagnation via large Cauchy jumps and chaotic re-initializations, yet achieves high-precision local search through Gaussian noise and random walk flights (Li, 21 Jul 2025). The UAV-path-planning paper emphasizes diversity curves, search-history plots, and the dynamic balance between exploration and exploitation, reporting that MISO maintains larger diversity early then focuses late (Li et al., 18 Jul 2025).

These accounts describe different mechanisms but converge on the same functional rationale: controlled diversity injection in the early search and progressively finer local refinement later. This suggests that the defining feature of MISO is not any single perturbation operator, but the deliberate orchestration of multiple operators across search phases.

6. Empirical results and application domains

In the UAV-path-planning paper, MISO is evaluated on 30 CEC2017 test functions and the CEC2022 test suite against 11 popular algorithms across different dimensions (Li et al., 18 Jul 2025). On CEC-2017 (30D), the reported Win–Tie–Loss is 18|12|0; at 50D and 100D, MISO leads with 25/30 and 25/30 wins. On CEC-2022 (20D), MISO wins 9/12 with no losses, and Friedman average ranks place it first across all dimensions. The paper also applies MISO to UAV 3D path planning with a cubic-spline waypoint model and total cost

$Q<0.25$7

where $Q<0.25$8 is path length, $Q<0.25$9 is altitude-deviation penalty, and $$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$0 is a smoothness term based on turning angles (Li et al., 18 Jul 2025). Over 30 runs, SO mean cost is reported as approximately 326.8 and MISO approximately 291.8, a decrease of 10.7%, with the best path smoother and having fewer sharp turns. The same paper further reports first-place Friedman ranking across six engineering design problems and lists representative best values such as welded beam design $$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$1, spring design $$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$2, cantilever beam design $$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$3, bearing design $$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$4, speed reducer design $$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$5, and three-bar truss design $$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$6 (Li et al., 18 Jul 2025).

In the trajectory-prediction paper, MISO is used to tune hyperparameters of a hybrid CNN-LSTM-Attention-AdaBoost neural network for medium- and long-term four-dimensional trajectory prediction (Li, 21 Jul 2025). The model applies the AdaBoost algorithm to divide multiple weak learners, and each submodel uses CNN to extract spatial features, LSTM to capture temporal features, and an attention mechanism to capture global features comprehensively. The strong learner model, combined with multiple sub-models, then optimizes the hyperparameters of the prediction model through the natural selection behavior pattern simulated by SO. The dataset contains 20 526 ADS-B records, comprising latitude, longitude, altitude, and time, from Xi’an to Tianjin during April–July 2024. Evaluation uses RMSE, MAPE, MAE, MAXAE, and $$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$7. The MISO-tuned model achieves RMSE = 125.43 m and MAPE = 1.3504%; this is 19.4% lower RMSE than the WOA-tuned model and 20.7% lower than the PSO-tuned model, while MAPE improves by up to 56.9% versus the GCO-tuned model (Li, 21 Jul 2025). On CEC2022 at $$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$8 over 12 functions, the same paper reports that MISO achieved the best average rank on 10/12 functions, that mean errors on unimodal and composition functions were 15–40% lower, and that Wilcoxon $$\begin{aligned} X_{i,m}(t+1)&=X_{\mathrm{rand},m}(t)\pm C_2\,A_m\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr],\ X_{i,f}(t+1)&=X_{\mathrm{rand},f}(t)\pm C_2\,A_f\,\bigl[(X_{\max}-X_{\min})\times\mathrm{rand}+X_{\min}\bigr], \end{aligned}$$9 in 10/12 cases versus PSO, WOA, and GWO (Li, 21 Jul 2025).

Ablation evidence is also reported. In the trajectory-prediction setting, adding MISO in the loss search reduced final loss by 39.89% over the AdaBoost baseline (Li, 21 Jul 2025). In the abstract of the same paper, the corresponding system is called SO-CLA-adaboost and is reported to outperform particle swarm, whale, and gray wolf in handling large-scale high-dimensional trajectory data (Li, 21 Jul 2025).

7. Interpretation, nomenclature, and open issues

A common misconception would be to treat MISO as a uniquely fixed optimizer. The 2025 literature does not support that reading. One paper defines MISO through adaptive random disturbance, adaptive Lévy flight, and elite leadership with Brownian motion (Li et al., 18 Jul 2025); another defines it through good-point-set initialization, adaptive thresholds, dual mutation, and composite flight (Li, 21 Jul 2025). The shared core is the retention of SO’s male/female behavioral structure and the insertion of multiple coordinated strategies around it.

Another important point concerns what the reported gains do and do not establish. The papers report strong benchmark rankings, lower errors, Wilcoxon significance results, and improved trajectory-prediction metrics in their respective testbeds (Li et al., 18 Jul 2025, Li, 21 Jul 2025). However, the UAV-path-planning paper explicitly notes that extra parameters require calibration and that the $C_2=0.05$0 switching design may be problem-dependent (Li et al., 18 Jul 2025). This suggests that MISO’s effectiveness depends not only on the choice of operators but also on schedule design and parameter control.

Future directions are stated explicitly only in the UAV-path-planning study. These include automated parameter control, applying MISO’s multi-strategy framework to other base optimizers, and extensions to dynamic, multi-objective, and large-scale real-time applications such as fault diagnosis, PV system parameter estimation, multi-agent task allocation, and point-cloud registration (Li et al., 18 Jul 2025). The trajectory-prediction study points in a complementary direction by embedding MISO into hyperparameter optimization for a hybrid CNN-LSTM-Attention-AdaBoost model, which suggests a broader role for MISO as a model-selection and training-configuration optimizer in sequence modeling and spatiotemporal prediction (Li, 21 Jul 2025).