Expert-Guided Memetic Walrus Optimizer

• The paper formulates adaptive curriculum sequencing as a multi-objective combinatorial optimization problem balancing concept coverage, time constraints, and learning-style compatibility.
• It employs an expert-guided, aging-weighted exploitation strategy combined with dual adaptive control signals to dynamically manage exploration and exploitation.
• A three-tier priority material sequencing mechanism ensures pedagogical validity by optimizing learning progression and adherence to prerequisite structures.

The Expert-Guided Memetic Walrus Optimizer (MWO) is a population-based metaheuristic designed for constrained, multi-objective optimization in adaptive curriculum sequencing (ACS) settings. MWO integrates memetic exploitation with domain-specific guidance, innovative adaptive exploration control, and a pedagogically informed sequence prioritization strategy to optimize learning material selection for personalized education scenarios.

1. Multi-Objective Problem Formulation

MWO formulates the ACS task as a multi-objective combinatorial optimization problem. The learning material selection for $T_s$ students from $T_m$ candidates is represented as $x \in \{0,1\}^{T_m}$. The objective function is a weighted sum over three pedagogically critical criteria: $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$ where:

• Concept Coverage/Redundancy:

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

with $\mathcal{R}$ as the union of concepts covered by selected materials, $\mathcal{E}$ as the required concepts. Penalty weight $\varepsilon_2 \gg \varepsilon_1$ prioritizes omission over redundancy.

• Time Constraint Violation:

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

• Learning-Style Compatibility:

$$\mathcal{O}_3 = \sum_{j=1}^{T_m} x_j \sum_{k=1}^4 |p_k - pm_{j,k}|$$

where $T_m$0, $T_m$1 are FSLSM-style coordinates of the student and material $T_m$2.

Experimental settings typically use $T_m$3, $T_m$4, $T_m$5, $T_m$6 (Huang et al., 16 Jun 2025).

2. Expert-Guided Search with Aging

A core innovation in MWO is the expert-guided, aging-weighted exploitation strategy. Unlike conventional metaheuristics, MWO maintains an expert pool and models each individual’s influence as a decaying exponential function of “age” (iterations since last being top-2). The update mechanism includes:

• Aging:
  • $T_m$7 each iteration; reset to zero if $T_m$8 is top-2.
  • Influence $T_m$9 if $x \in \{0,1\}^{T_m}$0, zero otherwise, with $x \in \{0,1\}^{T_m}$1.
• Guided Update:
  • For agent $x \in \{0,1\}^{T_m}$2, select expert $x \in \{0,1\}^{T_m}$3 (from $x \in \{0,1\}^{T_m}$4) with probability $x \in \{0,1\}^{T_m}$5.
  • Position update:

  $x \in \{0,1\}^{T_m}$6

This mechanism allows for informed, diversity-enhanced exploitation and mitigates stagnation in local optima (Huang et al., 16 Jun 2025).

3. Adaptive Control Signal Framework

MWO employs a dual "danger/safety" signal system for dynamically balancing exploration and exploitation:

• Danger Signal ($x \in \{0,1\}^{T_m}$7): Primarily active in early iterations, calculated as $x \in \{0,1\}^{T_m}$8, where $x \in \{0,1\}^{T_m}$9 and $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$0. High $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$1 triggers migration steps to preserve diversity.

• Safety Signal ($$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$2): Becomes prominent as the search progresses, defined by

$$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$3

with $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$4.

Update actions depend on ($$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$5, $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$6) values, modulating between migration, global search (with Halton sequences), local refinement, and best-case recombination to adjust search breadth as needed (Huang et al., 16 Jun 2025).

4. Three-Tier Priority Material Sequencing

Post-optimization, MWO introduces a three-tier mechanism for sequencing selected learning materials:

• High Priority: Materials fully covering required concepts and within estimated ability, score $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$7 (prerequisite-weighted).

• Medium Priority: Within ability but only partially covering concepts, $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$8, $$\min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x)$$9.

• Challenge: Materials slightly above current ability, $$\mathcal{O}_1 = \varepsilon_1 \bigl(|\mathcal{R}|-|\mathcal{R} \cap \mathcal{E}|\bigr) + \varepsilon_2 \bigl(|\mathcal{E}|-|\mathcal{R} \cap \mathcal{E}|\bigr)$$0, $$\mathcal{O}_1 = \varepsilon_1 \bigl(|\mathcal{R}|-|\mathcal{R} \cap \mathcal{E}|\bigr) + \varepsilon_2 \bigl(|\mathcal{E}|-|\mathcal{R} \cap \mathcal{E}|\bigr)$$1 (progress-indexed).

Final score: $$\mathcal{O}_1 = \varepsilon_1 \bigl(|\mathcal{R}|-|\mathcal{R} \cap \mathcal{E}|\bigr) + \varepsilon_2 \bigl(|\mathcal{E}|-|\mathcal{R} \cap \mathcal{E}|\bigr)$$2 with $$\mathcal{O}_1 = \varepsilon_1 \bigl(|\mathcal{R}|-|\mathcal{R} \cap \mathcal{E}|\bigr) + \varepsilon_2 \bigl(|\mathcal{E}|-|\mathcal{R} \cap \mathcal{E}|\bigr)$$3. Materials are sorted by descending $$\mathcal{O}_1 = \varepsilon_1 \bigl(|\mathcal{R}|-|\mathcal{R} \cap \mathcal{E}|\bigr) + \varepsilon_2 \bigl(|\mathcal{E}|-|\mathcal{R} \cap \mathcal{E}|\bigr)$$4, enforcing prerequisite graph constraints to yield the learning sequence. This structuring explicitly maintains pedagogical validity while optimizing for progression and prerequisite adherence (Huang et al., 16 Jun 2025).

5. Workflow and Computational Complexity

The overall MWO algorithm involves initializing a population of candidate solutions, iteratively updating positions based on the adaptive expert-guided scheme and control signals, continuously tracking best solutions, and generating the final learning sequence by the priority mechanism.

Abridged pseudocode structure:

$\mathcal{E}$6

Each iteration incurs $$\mathcal{O}_1 = \varepsilon_1 \bigl(|\mathcal{R}|-|\mathcal{R} \cap \mathcal{E}|\bigr) + \varepsilon_2 \bigl(|\mathcal{E}|-|\mathcal{R} \cap \mathcal{E}|\bigr)$$5 time, dominated by fitness evaluations ($$\mathcal{O}_1 = \varepsilon_1 \bigl(|\mathcal{R}|-|\mathcal{R} \cap \mathcal{E}|\bigr) + \varepsilon_2 \bigl(|\mathcal{E}|-|\mathcal{R} \cap \mathcal{E}|\bigr)$$6) and sequence sorting, yielding an overall time complexity of $$\mathcal{O}_1 = \varepsilon_1 \bigl(|\mathcal{R}|-|\mathcal{R} \cap \mathcal{E}|\bigr) + \varepsilon_2 \bigl(|\mathcal{E}|-|\mathcal{R} \cap \mathcal{E}|\bigr)$$7. Standard configuration: $$\mathcal{O}_1 = \varepsilon_1 \bigl(|\mathcal{R}|-|\mathcal{R} \cap \mathcal{E}|\bigr) + \varepsilon_2 \bigl(|\mathcal{E}|-|\mathcal{R} \cap \mathcal{E}|\bigr)$$8, $$\mathcal{O}_1 = \varepsilon_1 \bigl(|\mathcal{R}|-|\mathcal{R} \cap \mathcal{E}|\bigr) + \varepsilon_2 \bigl(|\mathcal{E}|-|\mathcal{R} \cap \mathcal{E}|\bigr)$$9, $\mathcal{R}$0, $\mathcal{R}$1, $\mathcal{R}$2 (Huang et al., 16 Jun 2025).

6. Experimental Evaluation

Comprehensive empirical evaluation is conducted on the OULAD dataset and standard continuous optimization benchmarks:

OULAD Results:

• Convergence Stability: Average fitness (±std, 30 runs):

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): Fraction of consecutive items with non-decreasing difficulty:

  • MWO: 95.3%
  • WO: 87.2%
  • SCSO: 85.1%
  • SOA: 83.4%
  • PEOA: 78.9%
• Sequence Quality (Student S1 Example):
  • MWO: Coverage 100.0%, DPR 90.7%, Alignment 100.0%; first 6 materials: 28→90→61→76→35→47
  • WO: Coverage 100.0%, DPR 83.3%, Alignment 96.7%; first 6 materials: 28→90→133→76→47→65
• Runtime: MWO 497 iterations / 150.6s; WO 495 iterations / 158.0s

Benchmark Functions:

• On $\mathcal{R}$3–$\mathcal{R}$4, MWO achieves objective $\mathcal{R}$5–$\mathcal{R}$6 (zero std).
• On TF$\mathcal{R}$7: $\mathcal{R}$8 versus optimum $\mathcal{R}$9.
• TF$\mathcal{E}$0 (Ackley) and TF$\mathcal{E}$1: zero std.
• TF$\mathcal{E}$2–$\mathcal{E}$3: matches or outperforms competitive methods with std $\mathcal{E}$4.
• Wilcoxon tests ($\mathcal{E}$5): MWO is statistically superior on 8/9 functions against SCSO/PEOA, 7/9 vs. SOA (Huang et al., 16 Jun 2025).

7. Synthesis and Significance

MWO's design incorporates (i) an expert-guided, aging-weighted search for robust and stable exploitation, (ii) a dual adaptive control signal framework for dynamic regulation of global/local search behaviors, and (iii) a pedagogically structured three-tier priority sequencing for curriculum validity. These synergistic elements yield state-of-the-art performance on ACS tasks—including high difficulty progression rates and convergence stability—while also exhibiting competitive robustness on standard global optimization benchmarks. This suggests applicability to a wider class of combinatorial and multi-objective optimization challenges, particularly where domain structure and constraint satisfaction are paramount (Huang et al., 16 Jun 2025).