Randomized Memetic ABC (RMABC) Algorithm

• The paper introduces RMABC, which integrates memetic strategies with randomized perturbations to enhance convergence and solution stability in multi-objective optimization.
• It employs expert-guided search mechanisms with an aging strategy and adaptive control signals to balance global exploration and local refinement.
• Empirical results on curriculum sequencing benchmarks demonstrate RMABC’s superior efficiency and improved optimization metrics compared to traditional methods.

The Expert-Guided Memetic Walrus Optimizer (MWO) is an advanced evolutionary optimization algorithm designed for Adaptive Curriculum Sequencing (ACS), a multi-objective problem situated in personalized online learning environments. MWO integrates an expert-guided strategy with an agent aging mechanism, a nonlinear adaptive control signal framework, and a hierarchical educational priority mechanism to produce high-quality, stable, and contextually meaningful curriculum sequences. Empirical validation on the OULAD dataset and standard optimization benchmarks establishes its superior optimization stability, curriculum relevance metrics, and generalization to complex multi-objective scenarios (Huang et al., 16 Jun 2025).

1. Multi-Objective Optimization Formulation

MWO formulates ACS as a multi-objective binary selection problem, where the selection of educational materials for $T_s$ students and $T_m$ materials is encoded as $x\in\{0,1\}^{T_m}$. For each student, the overall loss function is

$$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$

where:

• $\mathcal{O}_1$: concept coverage/redundancy, penalizing uncovered required concepts and minimally penalizing redundant coverage,

$$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$

with $\mathcal{R}=\bigcup_{j:x_j=1}Cm_j$, $\mathcal{E}=\bigcup_i C_i$.

• $\mathcal{O}_2$: time constraint violation,

$$\mathcal{O}_2 = \begin{cases} \varepsilon_3, & \sum_{j=1}^{T_m}x_j\,Ts_j \notin [\underline{T},\overline{T}], \ 0, & \text{otherwise}, \end{cases}$$

• $T_m$0: learning-style compatibility,

$T_m$1

where $T_m$2 is the $T_m$3-th FSLSM coordinate of the student, $T_m$4 that of material $T_m$5.

Typical hyperparameter settings: $T_m$6, $T_m$7, $T_m$8, $T_m$9. This combination strongly penalizes missing concepts, with lower cost for redundancy or time/style mismatches.

2. Expert-Guided Strategy and Aging Mechanism

MWO maintains an expert pool comprising all agents, assigning each an exponentially decaying influence weight based on age-in-population. For agent $x\in\{0,1\}^{T_m}$0: $x\in\{0,1\}^{T_m}$1

$x\in\{0,1\}^{T_m}$2

For updating, better-fitness experts $x\in\{0,1\}^{T_m}$3 are sampled with probability $x\in\{0,1\}^{T_m}$4, and the update is

$x\in\{0,1\}^{T_m}$5

This mechanism improves exploitation and avoids premature convergence by favouring recent successful search directions.

3. Adaptive Control Signal Framework

MWO employs two nonlinear control signals per iteration:

• Danger signal $x\in\{0,1\}^{T_m}$6: Promotes exploration in early search phases.

$x\in\{0,1\}^{T_m}$7

• Safety signal $x\in\{0,1\}^{T_m}$8: Gradually increases exploitation.

$x\in\{0,1\}^{T_m}$9

Update strategies: 1. If $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$0: Migration update.

$$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$1

1. Else if $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$2:
   • Top-$$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$3 ("males"): Halton-based global moves.
   • Others: Local search around $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$4.
2. Else: Combine best and second-best positions:

   $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$5

   $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$6

   $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$7

This regime facilitates a dynamic and data-driven balance between global exploration and local refinement.

4. Three-Tier Curriculum Sequencing and Priority Assignment

After selecting optimal material subsets, MWO applies a hierarchical sequencing:

• High priority ($$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$8): Full concept coverage, within student ability, prioritized by prerequisite strength,

$$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$9

• Medium priority ($\mathcal{O}_1$0): Partial coverage, matched to ability,

$\mathcal{O}_1$1

• Challenge set ($\mathcal{O}_1$2): Slightly above ability, promotes progress,

$\mathcal{O}_1$3

• Final sequence score:

$\mathcal{O}_1$4

Materials are sorted by descending $\mathcal{O}_1$5, ensuring prerequisite constraints.

5. Algorithmic Structure and Computational Complexity

MWO’s control flow follows the pseudocode:

$\mathcal{R}=\bigcup_{j:x_j=1}Cm_j$1

With recommended parameters: population $\mathcal{O}_1$6, iterations $\mathcal{O}_1$7, materials $\mathcal{O}_1$8. Per-iteration complexity is $\mathcal{O}_1$9, overall $$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$0, dominated by fitness computation and sequence sorting.

6. Empirical Validation and Performance Analysis

Experimental Results on OULAD

MWO demonstrates superior convergence stability: | Algorithm | Avg. fitness | Std. deviation | |-----------|--------------|----------------| | MWO | 598.48 | 18.02 | | WO | 641.62 | 28.29 | | SCSO | 814.85 | 329.43 | | SOA | 972.65 | 422.11 | | PEOA | 866.35 | 315.62 |

Difficulty Progression Rate (DPR), the fraction of consecutive curriculum materials with non-decreasing difficulty:

• MWO: $$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$1
• WO: $$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$2
• SCSO: $$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$3
• SOA: $$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$4
• PEOA: $$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$5

For a representative student (S1): | Algorithm | Coverage (%) | DPR (%) | Alignment (%) | First 6 Materials | |-----------|-------------|---------|--------------|-----------------------| | MWO | 100.0 | 90.7 | 100.0 | 28→90→61→76→35→47… | | WO | 100.0 | 83.3 | 96.7 | 28→90→133→76→47→65… |

MWO converges in $$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$6 iterations ($$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$7s runtime), WO in $$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$8 iterations ($$\mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|),$$9s).

Standard Benchmark Function Validation

MWO achieves high performance and statistical superiority (Wilcoxon test, $\mathcal{R}=\bigcup_{j:x_j=1}Cm_j$0) on 8/9 functions versus SCSO and PEOA, and on 7/9 versus SOA. Notably, MWO attains solutions with standard deviation near zero on unimodal functions, and matches or outperforms state-of-the-art on multimodal and hybrid benchmarks.

7. Synthesis and Impact

MWO synergizes three distinct innovations: (1) an expert-guided, aging-weighted search dynamic that improves exploitation while preventing premature convergence; (2) a nonlinear, data-driven adaptive control framework for exploration–exploitation trade-off; and (3) a multi-level, pedagogically motivated priority sequencing mechanism ensuring educational relevance and adherence to constraints.

The empirical findings demonstrate MWO’s efficacy in generating curriculum sequences with high concept coverage, realistic progression, learner-style alignment, and robust optimization stability. Its framework generalizes robustly to domain-agnostic optimization problems, suggesting potential for broader application in constrained multi-objective optimization domains where solution stability and nuanced prioritization are critical (Huang et al., 16 Jun 2025).