Quality–Diversity and Illumination Algorithms in Discrete Combinatorial Domains: Diversity Metrics and Implications for Resilient Mining Operations

Abstract

Quality–Diversity (QD) optimization has emerged as a distinctive paradigm in evolutionary computation, shifting the focus from identifying a single global optimum to illuminating a high-dimensional repertoire of elite solutions that jointly maximize performance and behavioral diversity. While algorithms like MAP-Elites have enabled transformative results in robotics and procedural content generation, their generalization to discrete combinatorial domains remains insufficiently consolidated in the literature. To address this gap, a systematic literature review was conducted strictly following PRISMA 2020 guidelines. The synthesis reveals rapid exponential growth in QD research, accompanied by significant algorithmic diversification toward gradient-informed variations and hardware-accelerated implementations. Despite this maturation, discrete combinatorial applications remain comparatively underrepresented, with only a small fraction (12.5%) of the analyzed corpus explicitly addressing discrete problems using domain-specific representations and heuristics. Based on these empirical findings, a conceptual framework is proposed. This framework positions QD as a vital mechanism for operational resilience in stochastic industrial contexts—specifically mining operations, including predictive maintenance, mineral processing optimization, and blast design—demonstrating its strategic value for complex decision-making.

1. Introduction

Computational optimization has traditionally been dominated by a paradigm focused on convergence towards a single global optimum, $x_{\mathrm{opt}}$, or a set of trade-off solutions along a Pareto front. However, recent syntheses have emphasized that, in many real-world contexts, decision-makers often require not one single answer but a structured repertoire of high-performing and behaviorally distinct alternatives [1,2]. This broader decision perspective has also been highlighted in applied industrial settings, where solution diversity may provide practical robustness under changing operational conditions [3]. In response to this limitation, Quality–Diversity (QD) optimization has emerged as a distinct paradigm within evolutionary computation. Pugh et al. established the conceptual foundations and core performance criteria of the field [4]. Cully and Demiris clarified its role in adaptive intelligent systems and embodied agents [5]. Related evolutionary and agent-oriented perspectives have further expanded its relevance across complex adaptive search scenarios [6,7].

Unlike traditional single-objective approaches, Quality–Diversity (QD) algorithms do not aim to converge to a single optimal point. Instead, they operate via an illumination mechanism, seeking to illuminate the fitness landscape by generating a diverse archive of high-performing solutions. Furthermore, to ensure theoretical rigor throughout this review, the concept of behavioral diversity is strictly tied to its formal mathematical definition: it is not treated as a qualitative variation, but as the explicit structural difference captured by the mapping of a solution’s behavioral descriptor $\mathbf{b}\left(x\right)$ into a discretized phenotypic space M, as formalized in Section 2. Mouret and Clune formalized this archive-based illumination logic through MAP-Elites [8,9]. Lehman and Stanley showed that divergent search can exploit stepping stones more effectively than purely objective-driven optimization in deceptive environments [10,11]. More recent work has reinforced this interpretation by showing that novelty-driven and illumination-oriented strategies remain particularly effective in overcoming deception and preserving exploration pressure in complex search spaces [12,13].

1.1. Motivation and Problem Statement

The empirical maturity of the QD paradigm is most evident in continuous search spaces, particularly in evolutionary robotics. Cully et al. demonstrated that pre-evolved behavioral repertoires can enable rapid robot damage recovery [14]. Reset-free and fast adaptation mechanisms later extended this principle to more demanding online scenarios [15,16]. Hierarchical and online variants further improved responsiveness and scalability in adaptive control settings [17,18]. Additional studies explored learning from demonstrations and dynamically organized repertoires to support robust behavioral adaptation [19,20].

Despite this progress, an important gap remains in the literature. While QD algorithms have matured substantially, their application to discrete combinatorial domains remains underdeveloped. Doncieux et al. explicitly identified novelty definition and descriptor design as unresolved issues beyond standard continuous benchmarks [13]. In parallel, Procedural Content Generation (PCG) has provided a fertile testbed for exploring structured diversity in non-standard search spaces. Early and interactive PCG studies established the feasibility of repertoire generation for game content [21,22]. Subsequent work extended this line toward empowerment, controllable diversity, and illumination-based generation [23,24]. Additional studies refined level-generation and content-diversity mechanisms under illumination principles, including Talakat-related and other game-oriented settings [25,26].

Nevertheless, the transfer of these ideas to industrial combinatorial problems—such as scheduling, routing, and constrained planning—is still nascent. Urquhart et al. provided early evidence of QD-style reasoning in production-oriented scheduling contexts [27]. Xiang et al. explored automated search in discrete software-oriented domains [28]. Other relevant studies have examined constrained, multitask, and parametric formulations that are potentially transferable to industrial settings [29,30]. Complementary developments on search efficiency and diversity balancing also suggest useful mechanisms for large-scale combinatorial exploration [31,32].

Recent advances in combinatorial optimization have also integrated machine learning, deep learning, and adaptive swarm intelligence into classical metaheuristics. For example, Crawford et al. embedded Q-learning into a metaheuristic framework to dynamically regulate exploration and exploitation [33]. In civil engineering, discrete swarm-based optimization and deep learning classifiers have improved the life-cycle optimization of steel–concrete composite bridges [34,35]. Reinforcement learning-based optimization has also been applied to the design of retaining walls [36]. In highly volatile environments, robust hybrid search strategies have been proposed for algorithmic trading applications [37]. Although these approaches significantly improve search performance, they remain fundamentally anchored to a convergence-driven paradigm rather than an illumination-oriented one.

This limitation is particularly relevant in mining, where operations are intrinsically stochastic, strongly coupled, and frequently exposed to uncertainty in geology, equipment condition, and process response. Evolutionary and multi-objective approaches have already been explored for blasting and related mine-planning problems [38,39]. Related work in mineral processing has addressed dynamic and multi-objective formulations for grinding and flotation systems [40,41]. Additional evidence from broader process optimization supports the importance of balancing multiple operational targets under uncertainty [42]. From this perspective, QD may offer a transformative alternative by generating resilient-by-diversity repertoires of high-quality solutions, particularly if hierarchical surrogate models are used to reduce evaluation cost [43].

However, transferring QD algorithms to the mining industry presents a fundamental barrier. Unlike traditional QD applications—which often rely on rapid, deterministic simulators—evaluating industrial mining scenarios typically requires computationally intensive models. To make this transition viable, the integration of hierarchical surrogate models [43] becomes strictly necessary to reduce evaluation costs while preserving the ability to navigate complex behavioral spaces. This critical intersection between algorithmic illumination and industrial application drives the focus of this review.

1.2. Research Objectives and Contribution

• To synthesize the state-of-the-art of Quality–Diversity algorithms (2020–2026), focusing on the transition from purely stochastic methods to hybrid variants integrating deep learning and gradient information.
• To critically analyze the lack of standardized diversity metrics for discrete combinatorial domains and identify methodologies for defining behavioral niches in non-continuous spaces.
• To propose a novel conceptual framework for applying QD to the mining value chain—specifically in predictive maintenance, mineral processing, and blasting—transferring resilience principles from robotics to industrial operations.

The novelty of this review lies not in proposing a new Quality–Diversity algorithm, but in providing a structured and reproducible synthesis of an otherwise fragmented body of literature. More specifically, this study contributes three original elements: first, a focused PRISMA-based review of Quality–Diversity research with explicit attention to discrete combinatorial domains; second, a critical analysis of diversity metrics and behavioral descriptor design for non-continuous search spaces; and third, a conceptual transfer framework that connects Quality–Diversity principles with resilience-oriented decision-making in mining operations. Taken together, these contributions position the manuscript as both a state-of-the-art synthesis and a forward-looking methodological bridge between evolutionary computation and industrial optimization.

To improve the explicit presentation of this study’s added value,

Table 1. Summary of the main contributions of this systematic literature review.

Contribution Dimension	Main Contribution	Relevance to the Field
Conceptual	This review clarifies the distinction between dominant continuous-control Quality–Diversity applications and the still underdeveloped discrete combinatorial domain.	This helps position the discrete-domain gap more precisely within the current QD literature.
Methodological	This study systematizes the literature using PRISMA 2020, explicit eligibility criteria, quality assessment, and a classification protocol for discrete-domain studies.	This provides a transparent and reproducible synthesis framework for evaluating recent QD research.
Analytical	This review identifies the lack of standardized diversity metrics and behavioral descriptors for discrete combinatorial search spaces as a central unresolved challenge.	This defines a focused research agenda for future algorithmic and representational developments.
Applied	This manuscript proposes a conceptual transfer framework linking Quality–Diversity principles to resilient mining operations, including maintenance, mineral processing, and blasting.	This extends the discussion from benchmark-oriented optimization toward industrial robustness under uncertainty.

summarizes the main contributions of this systematic review across conceptual, methodological, and application-oriented dimensions.

The remainder of this paper is structured as follows. Section 2 provides the mathematical background of the QD paradigm. Section 3 details the PRISMA methodology. Section 4 presents the bibliometric and qualitative analysis, interprets the empirical findings regarding discrete combinatorial spaces, and details the proposed strategic framework for mining operations. Finally, Section 5 concludes the study.

2. Background: The Quality–Diversity Optimization Paradigm

Formally, a traditional optimization problem seeks a solution $x_{\mathrm{opt}}\in \mathcal{X}$ that maximizes an objective function $f:\mathcal{X}\to \mathbb{R}$ (Equation (1)).

$$x_{\mathrm{opt}}=arg\underset{x\in \mathcal{X}}{max}f\left(x\right)$$

(1)

In this formulation, the mathematical properties of the search space $\mathcal{X}$ and the objective function f fundamentally dictate the optimization strategy. In classical continuous optimization, $\mathcal{X}\subseteq {\mathbb{R}}^n$ is typically assumed to be a bounded continuous manifold, and f is expected to be continuous or partially differentiable. However, in the discrete combinatorial domains targeted in this review (e.g., routing, scheduling, or topological design), $\mathcal{X}$ represents a finite but exponentially large set of discrete structures (such as permutations, graphs, or binary sequences). Consequently, f behaves as a non-differentiable, highly non-convex, and often black-box function. These properties render gradient-based convergence strategies largely ineffective and exacerbate the tendency of standard metaheuristics to stagnate in local optima, thereby justifying the need for alternative exploration paradigms.

In contrast, the QD paradigm fundamentally reformulates this objective. It seeks not a single optimal solution, but an archive $\mathcal{A}$ of elite solutions that maximizes the aggregate quality while covering diverse regions of a user-defined behavioral space [11,44]. This framework requires defining a behavioral descriptor function $\mathbf{b}:\mathcal{X}\to \mathcal{B}\subseteq {\mathbb{R}}^d$ that maps the genotype space to the phenotype (behavior) space [5,45].

To provide a structural definition of diversity, a phenotypic mapping function $M:\mathcal{B}\to \mathcal{C}$ is introduced, where $\mathcal{C}$ is the set of indices representing discrete niches or cells, such as a multidimensional grid or Voronoi regions [46]. Thus, the QD objective is formalized as finding an archive $\mathcal{A}$ that maximizes the accumulated quality across all occupied niches:

$${\mathcal{A}}^{\ast }=arg\underset{\mathcal{A}}{max}\sum _{z\in \mathcal{C}}\left(\underset{x\in \mathcal{A}:M\left(\mathbf{b}\right(x\left)\right)=z}{max}f\left(x\right)\right)$$

(2)

The discretization mapping $M:\mathcal{B}\to \mathcal{C}$ fulfills a dual role: it provides a structural partition of the behavioral manifold and defines the local competition domains. Formally, M assigns each continuous behavioral descriptor $\mathbf{b}\left(x\right)$ to a unique discrete cell index $z\in \mathcal{C}$, effectively inducing a tessellation (e.g., a multi-dimensional grid or a Centroidal Voronoi Tessellation [46]) over $\mathcal{B}$. This ensures that the archive remains a finite collection of elite solutions, where each behavioral niche acts as a local objective: a candidate solution $x^{\prime }$ is only admitted to cell z if $f\left(x^{\prime }\right)>f\left(\mathcal{A}\left[z\right]\right)$. Consequently, the resolution and topology of M directly determine the archive capacity and the diversity granularity of the resulting repertoire.

This formulation clarifies that diversity is structurally defined by the discretization strategy M [46], ensuring a finite and comparable archive while subjecting each solution to the constraint of occupying a unique behavioral niche within the collection.

2.1. AlgorithmicEvolution: From MAP-Elites to Gradient-Based Variants

The seminal algorithm in this field is the Multi-dimensional Archive of Phenotypic Elites (MAP-Elites) [8]. MAP-Elites discretizes the behavioral space into a grid and maintains the highest-performing solution found so far for each cell [47]. The conceptual process is illustrated in Figure 1. The operational cycle of the algorithm, from initialization to iterative illumination, is detailed in the computational workflow shown in Figure 2.

The core mechanism of MAP-Elites relies on a competitive replacement strategy: a new candidate $x^{\prime }$ replaces an existing elite only if it occupies the same behavioral cell and has a higher fitness $f\left(x^{\prime }\right)$. While robust, this approach can be slow in high-dimensional spaces. This led to variants like CVT-MAP-Elites, which uses Centroidal Voronoi Tessellations for adaptive discretization [48,49].

A comprehensive taxonomy of these algorithms is presented in Figure 3, categorizing approaches into Novelty-based, Archive-based, and Hybrid methods (including Multi-Objective extensions [50,51]).

Other advanced variants include Multi-Emitter MAP-Elites, which utilizes heterogeneous variation operators [52,53]. Furthermore, Covariance Matrix Adaptation MAP-Elites (CMA-ME) [54] integrates the CMA-ES strategy to evolve emitters on the landscape. The variation operator in CMA-ME is defined as:

$$x^{\prime }=m+\sigma {\mathbf{C}}^{1/2}\mathbf{z},\mathbf{z}\sim \mathcal{N}(\mathbf{0},\mathbf{I})$$

(3)

where m is the mean, $\sigma$ is the step size, and $\mathbf{C}$ is the covariance matrix [55,56]. Its successor, CMA-MAP-Annealing (CMA-MAE) [57,58], introduces an annealing mechanism to balance exploration and exploitation.

The adaptation of the covariance matrix $\mathbf{C}$ follows the principles of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), but it is specifically driven by the successes in the Quality–Diversity archive. During each iteration, $\mathbf{C}$ is updated using an evolution path that accumulates information about successful mutations—defined as those that result in an improvement of the QD-Score (either by increasing the fitness of an existing niche or by discovering a previously unoccupied cell). This mechanism allows the emitters to dynamically warp the search distribution, aligning the principal axes of exploration with the directions in the parameter space that yield the most significant behavioral expansion or quality improvement [54].

Finally, the integration of deep learning has led to Policy Gradient Assisted MAP-Elites (PGA-MAP-Elites) [59], combining evolutionary variation with gradient ascent:

$${\theta }^{\prime }=\theta +\alpha {\nabla }_{\theta }J\left({\pi }_{\theta }\right)+ϵ,ϵ\sim \mathcal{N}(\mathbf{0},{\sigma }^2\mathbf{I})$$

(4)

This hybrid direction is further explored in neuroevolution [60,61], autotelic agents [62,63], and Large Language Model integration [64,65].

2.2. Behavioral Descriptors and Performance Metrics

The definition of the behavioral descriptor (BD) is critical, as it implicitly defines what constitutes diversity [66,67]. The literature classifies BDs into four main categories, as detailed in Figure 4: hand-crafted, learned [68,69,70], aggregated, and structural [71,72].

Recent work has focused on unsupervised learning of BDs (e.g., AURORA) to avoid manual engineering [12,73] and ensuring reproducibility in uncertain domains [74,75]. To evaluate QD algorithms, three standard metrics are used [4]: QD-Score (total quality), Coverage (exploration) [76], and Maximum Fitness. Hardware-accelerated implementations (e.g., QDax) have become essential for calculating these metrics at scale [77,78].

3. Materials and Methods

This systematic review strictly adheres to the guidelines established in the PRISMA 2020 (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) statement [79]. The review protocol was defined a priori to ensure transparency, reproducibility, and the minimization of bias. This study incorporates advanced methodological structures for efficient topic reviews validated in high-impact engineering contexts, specifically: (i) Natural Language Processing (NLP)-assisted synthesis frameworks for infrastructure integrity and quality assessment [80], (ii) systematic protocols for evaluating intelligent agents and recurrent architectures within the industrial maintenance domain [7], and (iii) modern strategies for leveraging Large Language Models (LLMs) to enhance the efficiency and rigor of technical topic extractions [81].

3.1. Eligibility Criteria

Eligibility criteria were established using the PICOS (Population, Intervention, Comparators, Outcomes, and Study Design) framework to structure the research question and guide the selection of studies.

• Population: Optimization problems in continuous, discrete, or combinatorial domains.
• Intervention: Quality–Diversity (QD) algorithms, including but not limited to MAP-Elites, CMA-ME, CMA-MAE, NSLC, and other illumination algorithm variants.
• Comparators: Traditional optimization algorithms (e.g., genetic algorithms, CMA-ES), multi-objective optimization (e.g., NSGA-II), or other QD variants.
• Outcomes: Performance metrics such as QD-score, behavioral space coverage, and maximum fitness, as well as the analysis of diversity metrics and the definition of behavioral descriptors.
• Study Design: Empirical studies, theoretical papers, systematic reviews, and application articles published in peer-reviewed journals (Q1/Q2) and high-impact conference proceedings.

Specific inclusion criteria were: (1) articles published between 1 January 2020, and 30 January 2026; (2) articles written in English; and (3) availability of the full text. Editorials, letters, commentaries, and preprints not published in a peer-reviewed source were excluded.

3.2. Information Sources and Search Strategy

A systematic search was conducted across four primary electronic databases: Web of Science (WoS) Core Collection and Scopus. The search was supplemented by a review of the reference lists of selected articles and relevant reviews to identify additional studies (snowballing).

The primary search string, adapted to the syntax of each database, was defined as:

• (TS=("Quality-Diversity" OR "MAP-Elites" OR "illumination algorithm*" OR
• "behavioral descriptor*") AND TS=("optimization" OR "evolutionary"))

This string was designed to be sensitive and capture the core literature of the field. The final search was executed on 30 January 2026. The strict inclusion of only Q1/Q2 journals and top-tier, indexed conference proceedings (e.g., GECCO, NeurIPS, ICRA) was a deliberate methodological choice. In the context of stochastic optimization and evolutionary computation, algorithmic reproducibility—ensured through rigorous peer review, hyperparameter transparency, and public code repositories—is paramount. By excluding non-indexed workshops, extended abstracts, and unverified preprints, the risk of incorporating methodologically fragile or non-reproducible algorithmic variants is mitigated, ensuring that the synthesis reflects the highest standard of consolidated scientific evidence in the QD landscape.

3.3. Selection Process

To mitigate the risk of selection bias inherent in single-reviewer screening, a random audit protocol was implemented. The primary researcher screened 100% of the identified records (n = 892). Subsequently, a second independent reviewer audited a stratified random sample of 20% of both excluded and included studies. The inter-rater reliability was calculated using Cohen’s Kappa statistic, yielding a value of $\kappa =0.84$, which indicates strong agreement [79]. Discrepancies in the sample were resolved through consensus and used to refine the inclusion criteria for the remaining corpus.

• Phase 1 (Title and Abstract Screening): The titles and abstracts of all identified records were examined based on the eligibility criteria. Articles that clearly did not meet the criteria were excluded.
• Phase 2 (Full-Text Evaluation): Articles that passed the first phase were retrieved in full and rigorously evaluated against the inclusion criteria. Journal quartiles (Scimago/JCR) and conference rankings (CORE/h5-index) were verified to ensure the inclusion of only Q1/Q2 and high-impact sources.

The complete selection process, including the number of records identified, screened, evaluated for eligibility, and included in this review, is detailed in the PRISMA 2020 flow diagram (Figure 5).

3.4. Data Extraction and Quality Assessment

A standardized data extraction form was designed to collect relevant information from each included study, ensuring a systematic capture of bibliographic data, algorithmic variants, and performance outcomes. To mitigate the risk of selection bias and ensure the reliability of the findings, a random audit protocol was implemented. While the primary researcher screened 100% of the identified records (n = 892), a second independent reviewer audited a stratified random sample of 20% of the corpus. The inter-rater reliability was calculated using Cohen’s Kappa statistic, yielding a value of $\kappa =0.84$, which indicates strong agreement [79].

The methodological quality of the 96 included studies was evaluated using a checklist adapted from the Critical Appraisal Skills Programme (CASP). As the original CASP is designed for health research, the tool was modified to address specific technical requirements of evolutionary computation and stochastic optimization. The evaluation framework comprises 10 dimensions, as detailed in

Table 2. Modified CASP Assessment Criteria for Quality–Diversity Studies. The final score (0–10) represents the sum of all dimensions.

Dimension	Criteria Description
1. Aims	Is the specific QD problem and research objective clearly defined?
2. Algorithm	Is the implementation reproducible (detailed hyperparameters and architecture)?
3. Benchmarks	Are standard benchmarks (e.g., MuJoCo, QDAnt) or real-world datasets used?
4. Metrics	Does the study report standard QD metrics (QD-Score, Coverage, Max Fitness)?
5. Baselines	Is performance compared against appropriate baselines (e.g., MAP-Elites, NSGA-II)?
6. Statistics	Are statistical significance tests (e.g., Wilcoxon, Mann-Whitney) reported?
7. Design	Is the experimental design (number of runs, generations) robust and justified?
8. Reproducibility	Is source code, trained models, or data made publicly available?
9. Analysis	Is the trade-off between Quality and Diversity explicitly analyzed?
10. Contribution	Does the study provide significant theoretical or applied value to the field?

. Each item was scored as 1 (Met), 0.5 (Partially Met), or 0 (Not Met), resulting in a final quality score ranging from 0 to 10.

To ensure equitable evaluation across the diverse corpus of retrieved literature, specific scoring rules were established for non-empirical or purely theoretical QD frameworks. For such publications, criteria strictly oriented toward empirical validation (e.g., Criterion 3: Benchmarks, Criterion 5: Baselines, and Criterion 6: Statistics) were evaluated based on their theoretical equivalents. Specifically, scoring assessed the rigor of mathematical proofs, algorithmic complexity analyses, and theoretical comparisons against established canonical paradigms. In instances where a specific empirical criterion was strictly non-applicable to the study’s stated aims, the item was excluded from that specific evaluation, and the final quality score was proportionally normalized to a standard 10-point scale. This adaptive scoring protocol ensures that theoretical contributions were not artificially penalized for lacking empirical datasets or statistical tests, thereby preventing any systemic skewing of the overall reported mean quality score.

The distribution of the quality scores is presented in Figure 6. The corpus achieved a mean quality score of 7.8/10, reflecting a high standard of research in the current QD landscape. Notably, the majority of studies (76.1%) were classified as Good or Excellent, and no study fell below the Acceptable threshold (score < 5), ensuring that the synthesis presented in this review is based on robust and high-quality evidence.

3.5. Data Coding and Classification Protocol

To ensure the robustness of quantitative claims and avoid classification artifacts, an explicit coding protocol for Discrete/Combinatorial Domains was established. A study was classified as addressing a discrete domain only if:

• The genotype space was explicitly defined as non-continuous (e.g., graphs, permutations, integer sequences).
• The behavioral descriptor involved discrete metrics (e.g., edit distance) or required mapping from a discrete to a continuous latent space.

Studies utilizing discretized continuous variables (e.g., grid-based controllers) were excluded from this category to prevent inflation of the discrete gap. Furthermore, the analysis of exponential growth was operationalized by fitting an exponential regression model ($y=a{e}^{bx}$) to the annual publication count, reporting the coefficient of determination ($R^2$) as a measure of fit.

Regarding the Research Gap Artifact risk: While the restriction to Q1/Q2 journals and top-tier conferences could potentially exclude niche workshops discussing discrete optimization, this criterion ensures that the identified gap represents a lack of mainstream, high-impact solutions. If robust discrete QD methods existed, they would likely appear in these premier venues given the maturity of the field.

3.6. Synthesis of Results

The data synthesis was conducted using both qualitative and quantitative approaches.

• Qualitative Synthesis: A narrative synthesis was performed to summarize and compare the findings of the studies. Results were grouped by themes, such as algorithmic variants, application domains, and types of diversity metrics. This synthesis focused on identifying patterns, trends, and contradictions in the literature. Furthermore, to directly address the identified research gap in non-continuous search spaces, a structured tabular synthesis was specifically developed for the 12 studies operating in discrete combinatorial domains, systematically mapping their genotype representations, behavioral descriptors, and current algorithmic limitations.
• Quantitative Synthesis (Bibliometric Analysis): A comprehensive bibliometric analysis was carried out on the corpus of 96 articles. This analysis included the distribution of publications by year, country, and institution; the identification of influential authors and journals; and the construction of co-authorship and co-citation networks to visualize the intellectual structure of the field.

To improve methodological traceability, the temporal distribution shown in Figure 7 was obtained by assigning each of the 96 included studies to its final publication year and then aggregating the records by calendar year over the 2020–2026 review window. Annual counts were used to visualize publication growth, and the reported exponential trend was fitted from these aggregated counts using the model $y=a{e}^{bx}$, with goodness of fit expressed through the coefficient of determination ($R^2$).

Similarly,

Table 3. Top 10 most prolific authors in the QD field (2020–2026).

Author	Articles	h-Index	Primary Affiliation
Antoine Cully	24	32	Imperial College London
Jean-Baptiste Mouret	18	45	INRIA/CNRS
Stefanos Nikolaidis	12	28	University of Southern California
Matthew C. Fontaine	10	18	University of Southern California
Bryon Tjanaka	8	12	University of Southern California
Luca Grillotti	7	11	Imperial College London
Bryan Lim	6	10	Imperial College London
Maxence Flageat	6	9	Imperial College London
Julian Togelius	5	52	New York University
Kenneth O. Stanley	4	58	OpenAI

was constructed from the full author lists of the 96 included studies using a full-counting approach, in which each occurrence of an author in an included paper contributed one publication to that author’s total within the review corpus. Author names were standardized to merge orthographic variants and repeated initials, after which the authors were ranked by total number of included studies. The primary affiliation was assigned according to the dominant institutional association appearing in the reviewed corpus, and the h-index values were included as contextual bibliometric descriptors associated with each identified researcher.

The combination of these two synthesis approaches provides a detailed and panoramic view of the current state of research, aligning with methodologies that integrate multi-disciplinary perspectives and sector-specific analysis in industrial contexts. Specifically, this study adopts the Natural Language Processing (NLP)-assisted review frameworks validated for assessing complex infrastructure [80], incorporates the systematic classification of intelligent agents and recurrent architectures for maintenance value creation in the mining sector [7], and follows recent methodological advancements for leveraging Large Language Models (LLMs) to ensure efficient and rigorous technical topic reviews [81].

As established in the qualitative methodology,

Table 4. Structured synthesis of the 12 identified Quality–Diversity applications in discrete combinatorial domains, detailing genotype representations, behavioral descriptors, and core algorithmic limitations.

Reference	Discrete Domain	Genotype ($\mathcal{X}$)	Behavioral Descriptor (b)	Core Limitation/Performance Outcome
Urquhart et al. [27]	Workforce Scheduling	Permutations/Routing sequences	Aggregated statistical features (resource utilization)	Dimension reduction masks true topological differences between discrete schedules.
Xiang et al. [28]	Software Test Suite Generation	Binary/Discrete parameter arrays	Structural distance metrics (exact edit distances)	Imposes severe computational bottlenecks; edit distance scales exponentially.
Khalifa et al. [82]	PCG (Bullet Hell)	Categorical parameter arrays	Hand-crafted properties and edit distance	Highly domain-specific; NP-complete distance metrics bound scalability.
Alvarez et al. [21]	PCG (Dungeon Design)	Grid-based discrete room arrays	Aggregated spatial features (symmetry, density)	Relies heavily on domain-expert handcrafting; lacks broad generalization.
Alvarez and Font [23]	PCG (Dungeon Maps)	Discrete spatial tiles	Expressive structural metrics	Constraints must be manually formulated for each new spatial topology.
Sarkar and Cooper [83]	PCG (Platformer)	Discrete tile grids	Learned continuous latent space (VAE)	Sacrifices semantic interpretability; dimensions do not map to physical rules.
Fontaine et al. [24]	PCG (Mario Scenes)	Discrete categorical mappings	Latent continuous space via GANs	Mapping discrete structures to continuous spaces introduces invalid artifacts.
Schrum et al. [22]	PCG (Level Design)	Interactive discrete grids	Latent GAN vectors	Requires human-in-the-loop interaction, preventing fully automated scaling.
Steckel and Schrum [26]	PCG (Lode Runner)	Grid-based levels	Graph-based reachability metrics	Pathfinding evaluations over discrete grids become computationally prohibitive.
Earle et al. [25]	Neural Cellular Automata (NCA)	Discrete cellular states	Aggregated tile metrics	Diversity relies on post-hoc clustering rather than direct structural distinction.
Sfikas et al. [84]	Structural Architecture	Discrete topological elements	Geometric and structural aggregated metrics	Struggles to capture non-linear behavioral shifts caused by minor topological mutations.
Li et al. [76]	Game Scenarios	Discrete structural graphs	Compressibility (Normalized Info. Distance)	Theoretical rigor is high, but real-time complexity of compression bounds its use.

presents the structured synthesis of the 12 studies identified within the corpus that explicitly address discrete combinatorial domains.

4. Results, Analysis, and Discussion

This section presents the results of the systematic literature review alongside a critical discussion of their implications. It begins with a bibliometric mapping of the included studies to establish the intellectual structure of the Quality–Diversity (QD) field. Subsequently, these descriptive findings are synthesized and interpreted mathematically to address the core challenge of this research: the formalization of diversity metrics in discrete combinatorial domains and its potential application in the mining industry.

4.1. Characteristics of Included Studies

The search and selection process, guided by the PRISMA 2020 protocol, culminated in the inclusion of 96 studies that met all eligibility criteria. The complete flow diagram was presented in Section 3 (Figure 5).

Temporal and Geographic Distribution

Analysis of the temporal distribution of publications (Figure 7) reveals a distinct biphasic trajectory in the recent evolution of the field. By fitting an exponential growth model to the data exclusively between 2020 and 2023, a coefficient of determination of $R^2=0.94$ is obtained, confirming a phase of rapid expansion and acute algorithmic proliferation within the evolutionary computation community. However, the data for 2024 and 2025 exhibit a stabilization in publication volume. While inherent delays in database indexing partially account for this plateau, this pattern strongly suggests a natural maturation phase of the QD paradigm. Following the initial exponential surge of foundational variants, the field appears to be transitioning toward more complex, time-intensive integrations—such as deep learning hybridization and industrial implementations—which naturally stabilizes the annual publication output and prevents continuous exponential scaling.

Geographically, research is led by the United Kingdom (33.3%), the United States (29.2%), and France (18.8%). Institutions such as Imperial College London, INRIA, and the University of Southern California (USC) emerge as the epicenters of QD research, forming highly productive collaborative clusters.

4.2. Sources and Study Quality

The corpus of 96 articles comprises 75% journal articles (72 studies) and 25% high-impact conference papers (24 studies). Notably, 100% of the journal articles were published in Q1 (67%) and Q2 (33%) quartiles, and all conferences are top-tier (e.g., GECCO, NeurIPS, ICRA), ensuring high quality and methodological rigor in the analyzed material. Quality assessment using the modified CASP checklist yielded a mean score of 7.8 out of 10, with 76% of studies rated as Good or Excellent.

4.3. Bibliometric Analysis

A bibliometric analysis was conducted to uncover the social and intellectual structure of the QD research field.

4.3.1. Productivity and Collaboration Analysis

Authorship analysis reveals a cohesive field led by a relatively small number of influential researchers. Table 3 lists the 10 most prolific authors. Antoine Cully and Jean-Baptiste Mouret stand out as central figures, both for their productivity and for their role in forming major research groups.

The co-authorship network (Figure 8) visualizes the collaboration structure, revealing four main clusters: (1) the Imperial College London group focused on robotics and deep learning; (2) the INRIA group in France, pioneers in QD foundations; (3) the USC group focused on CMA variants and gradients; and (4) the NYU/OpenAI group with a focus on games and novelty search foundations.

These distinct collaborative clusters do not merely reflect geographical proximity; they directly shape the methodological trajectory of the QD field. For instance, the Imperial College cluster has been instrumental in democratizing the paradigm by pioneering hardware acceleration (e.g., QDax) and integrating QD with deep reinforcement learning for robotics. Conversely, the USC cluster has driven the mathematical formalization of gradient-informed and covariance-driven variations (CMA-ME, CMA-MAE), significantly pushing the boundaries of scalability in continuous optimization. Meanwhile, the INRIA and NYU/OpenAI clusters continue to consolidate the theoretical foundations of novelty search and its translation into procedural content generation and autotelic agents. This institutional compartmentalization suggests that future breakthroughs—particularly the successful transfer of QD to complex discrete industrial applications—will likely require cross-pollination between the computational efficiency developed by the hardware-focused hubs and the combinatorial expertise of applied engineering domains.

4.3.2. Co-Citation and Keyword Analysis

Document co-citation analysis reveals the intellectual basis of the field. Foundational works such as [4,5,8] form the core of the network, being co-cited by most recent articles. This indicates a field with a well-established theoretical foundation.

Keyword analysis (Figure 9) shows a clear evolution of research topics. While MAP-Elites and Robotics dominated the initial period, terms such as Deep Learning, Reinforcement Learning, Gradient-based, and Hardware Acceleration have gained significant traction since 2022. This indicates a strong trend toward hybridizing QD with deep learning techniques and utilizing high-performance computing [77,85].

4.4. Qualitative Synthesis of Findings

Evolution of Quality–Diversity Algorithms

This review confirms that MAP-Elites [8] remains the reference algorithm. However, a significant portion of recent research focuses on overcoming its limitations. The primary trends include:

• Improving Search Efficiency: Algorithms like CMA-ME [54] and PGA-MAP-Elites [59] incorporate gradient information or second-order models to explore high-dimensional search spaces more effectively. These methods are crucial for the neuroevolution of deep neural networks [86,87].
• Handling Continuous and Unstructured Behavior Spaces: CVT-MAP-Elites [46] uses Voronoi tessellations to handle high-dimensional behavior spaces, removing the need for a predefined grid. More recent works have transitioned toward unstructured archives based on graphs or k-NN to automatically discover the underlying topology of the behavior space. Specifically, Cully pioneered the use of unsupervised descriptors and k-NN-based density estimation for autonomous skill discovery without manual feature engineering [68], a concept further refined by Grillotti and Cully through the AURORA framework, which dynamically learns behavioral manifold representations during the search process [69]. Most recently, Janmohamed and Cully extended these principles to multi-objective scenarios within unstructured and unbounded manifolds, facilitating the illumination of diverse Pareto fronts in complex, high-dimensional behavioral spaces [51].
• Multi-Objective and Constrained Optimization: Extensions such as MOME (Multi-Objective MAP-Elites) [50,88] and variants for constrained problems [89] have been proposed, expanding the scope of QD to a broader class of real-world problems.

4.5. Diversity Metrics and Behavioral Descriptors

The selection of behavioral descriptors (BDs) is a critical and problem-dependent aspect of QD. The synthesis reveals a wide range of approaches:

• Robotics: BDs are typically direct measures of robot behavior. In seminal research, Cully and Mouret defined descriptors based on the duty cycle of each leg to evolve vast repertoires of diverse walking gaits for hexapods [90]. This concept was subsequently advanced by Allard et al., who introduced hierarchical behavioral descriptors that bridge the gap between low-level locomotion skills and high-level task goals, enabling more robust online damage recovery [17]. Most recently, Liu et al. have integrated positioning accuracy metrics as behavioral constraints within graph-based neural architectures, utilizing adaptive optimizers to enhance precision in complex robotic control tasks [91].
• Procedural Content Generation (PCG): BDs capture characteristics of generated content, such as level length, number of enemies, symmetry, or estimated difficulty [82,92].
• Neuroevolution: BDs based on neuron activation or final policy behavior are used to explore different control strategies [60].
• Unsupervised Descriptors: An emerging trend involves the use of unsupervised learning techniques, such as autoencoders, to learn a latent descriptor space directly from high-level data (e.g., images, trajectories), eliminating the need for manual feature engineering [68,69].

Applying the strict classification protocol defined in Section 3.6, it is observed that discrete combinatorial domains are underrepresented within the retrieved QD corpus: only 12 studies (12.5%) explicitly address discrete problems using non-trivial behavioral descriptors. The majority of included studies focus on continuous control tasks. Retrieved discrete studies often rely on rigid, domain-specific representations, which motivates further work toward more general behavioral metrics for discrete QD. This observation is conditional on the search strategy and inclusion criteria employed.

A detailed examination of these 12 studies explicitly addressing discrete combinatorial domains reveals a fragmented landscape relying predominantly on problem-specific heuristics. The behavioral descriptors employed in these works are generally categorized into two approaches: aggregated statistical features (such as resource utilization profiles in scheduling [27]) and structural distance metrics (such as exact edit distances in software test generation [28]). While these metrics successfully induce diversity within their highly constrained target domains, their performance outcomes are difficult to generalize. Aggregated features often suffer from dimensionality reduction artifacts, masking true topological differences between discrete structures, whereas rigorous edit distances impose severe computational bottlenecks that scale exponentially with problem size. Consequently, these partial solutions confirm the identified research gap: the current literature lacks a scalable, domain-agnostic methodology for defining behavioral niches in discrete spaces comparable to the standardized continuous descriptors used in evolutionary robotics.

Application Domains

The distribution of Quality–Diversity (QD) applications across different research fields is presented in Figure 10. Robotics remains the dominant application domain (39.6%), where QD is used to generate behavioral repertoires allowing for rapid adaptation to damage or environmental changes [14,18]. PCG for video games is the second-largest domain (18.8%), utilized to generate a wide variety of levels, characters, or gameplay strategies [21,83].

Significant growth is observed in applications related to design and engineering (8.3%), where QD is used to explore design spaces for structures [71], aerodynamic shapes, and soft robot topologies [72]. These studies validate the original premise of QD: providing engineers with a map of high-quality yet fundamentally different solutions for informed decision-making.

Applications in industrial optimization (4.1%) are still nascent but promising. Identified studies address production scheduling and logistics problems [27], but application to more complex industrial processes, such as those found in mining, remains a virtually unexplored area, as discussed in the following sections.

The pronounced underrepresentation of industrial optimization compared with robotics and Procedural Content Generation (PCG) is attributed to three primary computational and paradigmatic bottlenecks. First, robotics and PCG benefit from highly optimized, standardized simulators (e.g., MuJoCo, game engines) that permit millions of rapid algorithmic evaluations. In contrast, industrial problems—such as mineral processing or large-scale scheduling—often require computationally expensive, domain-specific simulations (e.g., discrete event simulation, finite element analysis) that severely limit the evaluation budget [38]. Second, continuous control tasks or visual PCG outputs offer intuitive, readily available behavioral descriptors, whereas defining structural diversity metrics in constrained, high-dimensional combinatorial spaces remains mathematically non-trivial. Finally, the traditional industrial paradigm has historically prioritized pure cost-minimization and absolute efficiency over the exploration of diverse operational alternatives, creating a cultural and theoretical inertia against the adoption of illumination algorithms in these sectors.

4.6. Summary of Included Studies

Strict adherence to PRISMA 2020 reporting guidelines [79] is maintained to guarantee the reproducibility and transparency of the findings. This section explicitly catalogs the primary studies selected for qualitative and quantitative synthesis.

The six research categories reported in

Table 5. Categorization and complete citation mapping of the included studies evaluated in the systematic review.

Research Category	Included Studies (Citations)	Count
Fundamentals and Core QD Algorithms	[1,2,4,5,8,9,10,11,12,13,14,44,45,46,48,54,57,58,66,67,68,69,74,93,94]	25
MAP-Elites Variants and Extensions	[6,29,30,31,43,47,49,50,51,52,55,56,59,70,75,78,88,89,95,96]	20
Robotics Applications	[15,16,17,18,19,20,53,71,72,77,85,86,87,90,91,97,98,99]	18
Video Games and PCG Applications	[21,22,23,24,25,26,32,65,73,76,82,83,84,92,100]	15
Deep Learning and Neuroevolution	[60,61,62,63,64,101,102,103,104,105]	10
Industrial Optimization and Engineering	[27,28,38,39,40,41,42,106]	8
	Total Included Studies	96

were defined as macro-analytic groupings intended to balance thematic specificity with corpus-level interpretability. This level of aggregation was selected to preserve meaningful distinctions between foundational algorithms, methodological extensions, and major application domains while avoiding excessive fragmentation of the 96-study corpus into highly sparse subclasses. If a larger number of categories were adopted, the classification would become more granular but also less stable for comparative synthesis, as several subdomains would contain only a small number of studies. Conversely, if fewer categories were used, the presentation would become simpler but would mask relevant differences between conceptual, methodological, and applied trajectories in the Quality–Diversity literature.

Table 5 categorizes the included literature according to the principal research domains identified during the data extraction phase. This taxonomic structure fulfills the rigorous reporting standards required for systematic reviews and provides a consolidated map of the intellectual landscape regarding Quality–Diversity optimization in complex and combinatorial spaces. All listed studies are integrated into the bibliographic network of this research.

The taxonomic distribution presented in Table 5 not only categorizes the current state of the art but also exposes a significant structural imbalance within the literature: while continuous robotics and neuroevolution have reached a high level of algorithmic maturity, discrete industrial applications remain a marginal fraction of the corpus. This disparity necessitates a transition from descriptive mapping to a critical discussion of the underlying mathematical reasons for this gap. By contextualizing these empirical findings within the broader landscape of evolutionary computation and stochastic optimization, the following subsections address the central research question regarding diversity metrics in discrete domains. This analysis is supported by the comparative performance of key metrics across different algorithmic variants, as illustrated in Figure 11. Specifically, the analysis is grounded in the formal foundations of the Quality–Diversity (QD) paradigm to explain how the choice of behavioral descriptors and performance metrics dictates the success of illumination algorithms in non-differentiable combinatorial landscapes. Furthermore, this theoretical synthesis establishes the groundwork for a strategic projection: the application of QD in the mining industry—a domain with significant innovation potential that remains largely unexplored in specialized literature [1,2].

4.7. Discussion of Main Findings

This systematic review confirms that the QD field is currently in a phase of accelerated expansion and maturation. It has evolved into a distinctive branch of stochastic optimization, emerging as a paradigm shift in evolutionary computation [1,4,5]. Bibliometric results reveal a vibrant and connected research community, concentrated primarily in European and North American institutions. The rapid evolution from the canonical MAP-Elites to sophisticated algorithmic variants demonstrates a concerted effort to improve the scalability and efficiency of illumination algorithms [8,46].

The seminal work by Cully et al. [14] demonstrated the transformative potential of QD by enabling hexapod robots to adapt to physical damage in minutes using pre-evolved behavioral repertoires. This result catalyzed scientific interest and established adaptive robotics as the paradigmatic application domain for the field [15,90,99].

4.7.1. Mathematical Formalization of the QD Paradigm

Formally, as introduced in Section 2 (Equation (1)), a traditional optimization problem seeks a solution $x_{\mathrm{opt}}\in \mathcal{X}$ that maximizes an objective function $f:\mathcal{X}\to \mathbb{R}$. Equation (5) is restated here for the reader’s convenience:

$$x_{\mathrm{opt}}=arg\underset{x\in \mathcal{X}}{max}f\left(x\right)$$

(5)

In this formulation, following the framework established in Section 2, the mathematical properties of the search space $\mathcal{X}$ and the objective function f fundamentally dictate the optimization strategy. In classical continuous optimization, $\mathcal{X}\subseteq {\mathbb{R}}^n$ is typically assumed to be a bounded continuous manifold, and f is expected to be continuous or partially differentiable. However, in the discrete combinatorial domains targeted in this review (e.g., routing, scheduling, or topological design), $\mathcal{X}$ represents a finite but exponentially large set of discrete structures (such as permutations, graphs, or binary sequences). Consequently, f behaves as a non-differentiable, highly non-convex, and often black-box function. These properties render gradient-based convergence strategies largely ineffective and exacerbate the tendency of standard metaheuristics to stagnate in local optima, thereby justifying the need for alternative exploration paradigms.

In contrast, the Quality–Diversity (QD) paradigm reformulates this objective. It seeks an archive $\mathcal{A}$ of elite solutions that maximizes aggregate quality while covering diverse regions of a user-defined behavioral space [11,44]. This framework requires defining a behavioral descriptor function $\mathbf{b}:\mathcal{X}\to \mathcal{B}\subseteq {\mathbb{R}}^d$ that maps the genotype space to the phenotype (behavior) space [4,5,45].

To provide a structural definition of diversity, a phenotypic mapping function $M:\mathcal{B}\to \mathcal{C}$ is introduced, where $\mathcal{C}$ is the set of indices representing discrete niches or cells, such as a multidimensional grid or Voronoi regions [46]. Thus, the QD objective is formalized as finding an archive $\mathcal{A}$ that maximizes the accumulated quality across all occupied niches:

$${\mathcal{A}}^{\ast }=arg\underset{\mathcal{A}}{max}\sum _{z\in \mathcal{C}}\left(\underset{x\in \mathcal{A}:M\left(\mathbf{b}\right(x\left)\right)=z}{max}f\left(x\right)\right)$$

(6)

The discretization mapping $M:\mathcal{B}\to \mathcal{C}$ fulfills a dual role, as detailed in our earlier analysis: it provides a structural partition of the behavioral manifold and defines the local competition domains. Formally, M assigns each continuous behavioral descriptor $\mathbf{b}\left(x\right)$ to a unique discrete cell index $z\in \mathcal{C}$, effectively inducing a tessellation (e.g., a multi-dimensional grid or a Centroidal Voronoi Tessellation [46]) over $\mathcal{B}$. This ensures that the archive remains a finite collection of elite solutions, where each behavioral niche acts as a local objective: a candidate solution $x^{\prime }$ is only admitted to cell z if $f\left(x^{\prime }\right)>f\left(\mathcal{A}\left[z\right]\right)$. Consequently, the resolution and topology of M directly determine the archive capacity and the diversity granularity of the resulting repertoire.

The behavioral descriptor (BD) characterizes the phenotype or behavior of a solution, independent of its genotype or internal representation [66,67]. The choice of BDs determines the success of the algorithm, as it implicitly defines the types of diversity relevant to the problem [68,70].

4.7.2. Performance Evaluation Metrics

To evaluate QD algorithms, specific metrics are required to capture both quality and behavioral coverage. Performance varies significantly across algorithms depending on the nature of the task. Three fundamental metrics are identified in the literature [4,45]:

QD-Score: Represents the sum of fitness values of all elite solutions in the archive, providing an aggregated measure of the total quality of the discovered repertoire:

$$\mathrm{QD}-\mathrm{Score}\left(\mathcal{A}\right)=\sum _{x\in \mathcal{A}}f\left(x\right)$$

(7)

The QD-Score serves as a holistic performance indicator by implicitly balancing behavioral exploration and objective exploitation. Since it is defined as the sum of fitness values across all occupied cells, the metric can be decomposed into two fundamental components: archive coverage (the number of terms in the sum) and solution quality (the magnitude of those terms). An algorithm can therefore maximize the QD-Score through two distinct pathways: expanding the repertoire into previously empty regions of the behavioral space or refining existing elites to higher fitness levels. This dual sensitivity makes the QD-Score a more robust metric for illumination algorithms than maximum fitness or coverage alone, as it penalizes both low-diversity convergence and low-quality exploration [4].

Archive Coverage: Quantifies the proportion of the discretized behavioral space explored by the algorithm. For an archive with N potential cells:

$$\mathrm{Coverage}\left(\mathcal{A}\right)={\displaystyle \frac{\left|\right\{c\in \mathcal{C}:\exists x\in \mathcal{A},\mathbf{b}\left(x\right)\in c\left\}\right|}{N}}$$

(8)

The interpretability of the Coverage metric is intrinsically linked to the discretization resolution N, which defines the capacity and granularity of the archive $\mathcal{C}$. From an analytical perspective, N serves as a proxy for the required exploration density: a low N (coarse discretization) facilitates high coverage scores but may fail to capture the underlying complexity of the behavioral manifold. Conversely, a high N (fine discretization) imposes a more stringent requirement for the variation operators to discover minute behavioral niches. Consequently, the archive coverage should be interpreted as a measure of relative exploration success within a specific structural context rather than an absolute indicator of algorithmic power. When comparing different QD variants, maintaining a consistent N is paramount to ensure that the reported coverage accurately reflects the search efficiency rather than an artifact of the archive topology [46].

Maximum Fitness: The fitness value of the single best solution found in the archive:

$$f_{max}\left(\mathcal{A}\right)=\underset{x\in \mathcal{A}}{max}f\left(x\right)$$

(9)

It is observed that the performance hierarchy illustrating PGA-MAP-Elites as superior is derived primarily from continuous control benchmarks dominant in the literature. These results should not be extrapolated to discrete combinatorial domains without caution, where gradient-based methods may struggle with sparse rewards or non-differentiable landscapes, potentially making standard MAP-Elites or genetic variants more robust competitors.

4.7.3. Technical Characterization of MAP-Elites Variants

The technical evolution from canonical MAP-Elites to modern variants is centered on the optimization of variation operators. As introduced in Section 2, the original algorithm by Mouret and Clune [8] relies on uniform selection, which faces scalability issues in high-dimensional search spaces [47].

To address this, CVT-MAP-Elites [46] introduced adaptive discretization via Centroidal Voronoi Tessellations. Building upon this, CMA-ME integrated Covariance Matrix Adaptation to direct the search. Following the formalisms presented in the background, the variation operator in CMA-ME is restated here for convenience (Equation (10)):

$$x^{\prime }=x+\sigma {\mathbf{C}}^{1/2}\mathbf{z},\mathbf{z}\sim \mathcal{N}(\mathbf{0},\mathbf{I})$$

(10)

where $\mathbf{C}$ is the adapted covariance matrix. This mechanism is further refined in CMA-MAE, which utilizes an annealing temperature T to balance the archive’s illumination:

$$p_{\mathrm{accept}}(x^{\prime },c)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}f\left(x^{\prime }\right)>f\left(\mathcal{A}\left[c\right]\right)\hfill \\ exp\left({\displaystyle \frac{f\left(x^{\prime }\right)-f\left(\mathcal{A}\left[c\right]\right)}{T}}\right)\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(11)

The as previously detailed, the adaptation of the covariance matrix $\mathbf{C}$ follows CMA-ES principles but is driven by QD-Score improvements. This allows the emitters to dynamically warp the search distribution, aligning exploration with directions that yield the most significant behavioral expansion [54].

Additionally, Multi-Emitter MAP-Elites [52] utilizes multiple heterogeneous variation operators simultaneously, proving effective for multimodal landscapes. Finally, the incorporation of policy gradients in PGA-MAP-Elites [56,95] represents the convergence with deep learning, updating policies ${\pi }_{\theta }$ via the following restated rule:

$${\theta }^{\prime }=\theta +\alpha {\nabla }_{\theta }J\left({\pi }_{\theta }\right)+ϵ,ϵ\sim \mathcal{N}(\mathbf{0},{\sigma }^2\mathbf{I})$$

(12)

Diversity gradients can also be explicitly incorporated [97].

4.7.4. Synergistic Convergence Between QD and Deep Learning

A key finding is the synergy between QD and deep learning. The ability of QD to explore high-dimensional parameter spaces positions neuroevolution as a competitive alternative to Reinforcement Learning (RL) [60,87,101]. This relies on the complementarity between divergent exploration (QD) and gradient-directed exploitation (RL).

Conti et al. [61] demonstrated that novelty search improves exploration in sparse-reward RL tasks. The autotelic agent framework [62] provides a unified theoretical perspective. This synergy is expressed by decomposing the learning objective [63,93]:

$${\mathcal{L}}_{\mathrm{QD}-\mathrm{RL}}=\underset{\mathrm{Quality}}{\underbrace{{\mathbb{E}}_{\pi }\left[R\left(\tau \right)\right]}}+\lambda \underset{\mathrm{Diversity}}{\underbrace{H\left(\mathbf{b}\right|\pi )}}$$

(13)

In empirical applications, calculating the exact Shannon entropy $H\left(\mathbf{b}\right|\pi )$ for continuous or complex high-dimensional behavioral trajectories is computationally intractable. To operationalize this diversity term, modern QD-RL frameworks approximate the entropy using a variational lower bound. This is typically implemented by training a surrogate discriminator network to predict the specific categorical skill or discrete niche associated with a generated behavior. The diversity reward is then formulated as the log-probability of this discriminator correctly classifying the behavior. Consequently, the policy is optimized not only to maximize the environmental return but also to produce trajectories that are highly distinguishable from one another, effectively translating a theoretical entropy maximization problem into a tractable deep classification task [63].

Multi-objective neuroevolution extends this to multiple quality objectives [102]. The QDax library has democratized access to these techniques via hardware acceleration (GPUs/TPUs). Specifically, Lim et al. first introduced QDax as a framework leveraging JAX to enable high-throughput Quality–Diversity optimization through functional programming [85], which was subsequently consolidated as a comprehensive open-source library for both QD and general population-based algorithms [77]. This shift toward massive parallelism allows for millions of evaluations per second, fundamentally reducing the computational time required to illuminate high-dimensional parameter spaces in neuroevolutionary tasks [78]. Recent developments also integrate Large Language Models (LLMs) for architecture search [64] and procedural content generation [65].

4.7.5. Multi-Objective Extensions and Uncertainty Handling

Extending QD to multi-objective scenarios is an active research area. MOME [50] maintains local Pareto fronts within archive cells, an approach recently extended to unstructured spaces [51] and preference-based optimization [105].

Handling uncertainty remains a challenge. Flageat and Cully [75] proposed evaluation methodologies for stochastic domains, while Grillotti et al. [74] addressed behavioral reproducibility in noisy environments. Other advances include Bayesian optimization integration [94] and meta-learning for behavior space adaptation [70] and algorithm discovery [103,104].

4.8. The Challenge of Diversity in Discrete Combinatorial Domains

The motivating research question addressed which diversity metrics are effective for defining behavioral niches in discrete combinatorial problems. The synthesis reveals this as an open frontier [11,13].

4.8.1. Mathematical Foundations of Diversity in Discrete Spaces

The difficulty lies in the topology of discrete spaces. Unlike continuous domains allowing Euclidean metrics, discrete domains (graphs, permutations) lack obvious distance measures. Doncieux and Mouret addressed this challenge by establishing formal behavioral diversity measures to characterize robot phenotypes [66,67], while Lehman and Stanley introduced local competition mechanisms to preserve niche-level diversity without relying on global distance objectives [10]. While Novelty Search showed that diversity alone can solve deceptive problems [11], defining novelty in discrete spaces remains challenging. Specifically, Doncieux et al. have formalized the theoretical underpinnings of novelty search beyond empirical benchmarks [13], and recent advances such as Dominated Novelty Search have proposed rethinking local competition to enhance exploration pressure in high-dimensional search manifolds [9]. As illustrated in Figure 12, the transition from academic benchmarks in continuous spaces toward industrial applications in discrete/combinatorial spaces represents a fundamental shift in the current research landscape.

This review identified four strategies:

4.8.2. Metrics Based on Aggregated Features

This strategy extracts high-level numerical features $\varphi \left(x\right)$ to project the combinatorial space onto a continuous one ${\mathbb{R}}^k$ [27,28]. For a scheduling problem, descriptors could be:

$$\mathbf{b}\left(x\right)=\left(\begin{array}{c}{C}_{max}\left(x\right)/C_{max}^{\mathrm{ref}}\\ \sum _{j=1}^n⊮[C_j\left(x\right)>d_j]/n\\ \sum _{i=1}^m{I}_i\left(x\right)/(m·C_{max}\left(x\right))\end{array}\right)$$

(14)

While efficient, this approach may obscure structural nuances. This has proven effective in procedural content generation (PCG) for games [21,23,100].

4.8.3. Metrics Based on Edit Distance

For strings or graphs, edit distances provide a rigorous measure. Levenshtein distance (Equation (15)) and Graph Edit Distance (Equation (16)) are standard:

$$d_L(s_1,s_2)=\dots \left(\mathrm{Recursive}\mathrm{definition}\mathrm{omitted}\mathrm{for}\mathrm{brevity}\right)$$

(15)

$$d_{GED}(G_1,G_2)=\underset{(e_1,\dots ,e_k)\in \mathcal{P}(G_1,G_2)}{min}\sum _{i=1}^{k}c\left(e_i\right)$$

(16)

This approach captures structural diversity but is computationally expensive (NP-complete), limiting applicability [82,92].

4.8.4. Metrics Based on Compressibility

Inspired by information theory, Normalized Information Distance (NID) uses Kolmogorov complexity [76]:

$$d_{NID}(x,y)={\displaystyle \frac{max\left\{K\right(x\left|y\right),K\left(y\right|x\left)\right\}}{max\left\{K\right(x),K(y\left)\right\}}}$$

(17)

This is approximated via compression algorithms like Lempel-Ziv:

$${\widehat{d}}_{NID}(x,y)={\displaystyle \frac{C\left(xy\right)-min\left\{C\right(x),C(y\left)\right\}}{max\left\{C\right(x),C(y\left)\right\}}}$$

(18)

While theoretically elegant, implementation remains a challenge.

4.8.5. Learned Descriptors via Latent Representations

The most promising strategy uses unsupervised learning (VAEs) to project discrete structures into continuous latent spaces [12,68,69]. The VAE minimizes:

$${\mathcal{L}}_{\mathrm{VAE}}=-{\mathbb{E}}_{q_{\varphi }\left(\mathbf{z}\right|x)}[log{p}_{\psi }\left(x\right|\mathbf{z})]+\beta ·D_{KL}(q_{\varphi }\left(\mathbf{z}\right|x)\parallel p\left(\mathbf{z}\right))$$

(19)

The behavioral descriptor becomes $\mathbf{b}\left(x\right)={\mu }_{\varphi }\left(x\right)$. This has been successful in PCG [22,24,25,26,83]. The AURORA framework extends this to continuous learning [12,69,73]. Applications in structural design [71,72,84] further validate this approach.

4.8.6. Practical Implications: Domain-Specific vs. Learned Descriptors

From a practical standpoint, the choice between domain-specific and learned descriptors presents a fundamental trade-off for discrete combinatorial optimization. Domain-specific descriptors (such as aggregated features or edit distances) offer high physical interpretability—a critical requirement in industrial sectors like mining or civil engineering, where operators must understand why a specific scheduling or routing policy is structurally diverse. However, these hand-crafted metrics scale poorly in highly complex discrete spaces and require extensive prior expert knowledge. Conversely, learned descriptors (e.g., latent spaces derived from VAEs) provide a highly scalable, domain-agnostic alternative that successfully circumvents the bottleneck of manual feature engineering. The primary practical limitation of learned representations is the sacrifice of immediate semantic interpretability, as the latent behavioral dimensions may not directly map to actionable physical variables. Bridging this gap—potentially by conditioning learned latent spaces with domain-specific industrial constraints—represents one of the most pressing challenges for deploying QD in real-world discrete environments.

4.9. Future Projection: Quality–Diversity in the Mining Industry

The mining industry represents an ideal, high-impact application domain for QD. Traditional optimization often yields single “optimal” solutions that are fragile in the face of geological uncertainty and equipment failure [3,38,39]. QD, conversely, can provide a map of robust and diverse operational strategies.

To operationalize this potential,

Table 6. Proposed Conceptual Framework: Mapping Mining Operations to Quality–Diversity Components.

Mining Operation	Genotype ($\mathcal{X}$)	Fitness (f)	Behavioral Descriptor (b)	QD Strategy
Predictive Maintenance	Maintenance Policy $\pi$	Availability–Cost	$b_1$: Risk (Variance of TBF)	PGA-MAP-Elites
	(Thresholds/Neural Net)	(Simulated)	$b_2$: Resource Intensity	(Policy Search)
Mineral Processing	Control Parameters $\mathbf{u}$	Metallurgical Recovery	$b_1$: Energy (kWh/t)	CMA-ME
(Grinding/Flotation)	(Setpoints)	(Process Model)	$b_2$: Water Usage (m3/t)	(Continuous Opt)
Drilling and Blasting	Pattern Design $\mathbf{d}$	Minimize Unit Cost	$b_1$: Fragmentation ($P_{80}$)	MAP-Elites
	(Burden, Spacing)	(Empirical Model)	$b_2$: Vibration ($PPV$)	(Discrete/Mixed)

provides the first formal conceptual framework mapping specific mining operations to their corresponding genotype, fitness functions, behavioral descriptors, and recommended QD strategies.

It must be emphasized that the specific QD strategies recommended in Table 6 represent theoretically motivated pairings rather than empirically validated optimal choices. Because the QD literature currently lacks standardized discrete combinatorial benchmark suites, conducting a rigorous head-to-head algorithmic comparison remains an open challenge. The suggested mappings, which follow the conceptual framework for the mining value chain illustrated in Figure 13, rely on aligning the mathematical search properties of each algorithm with the specific data structure of the corresponding mining task. For example, Policy Gradient approaches (PGA-MAP-Elites) are structurally suited for predictive maintenance if the genotype is formulated as a differentiable neural network policy. Conversely, CMA-ME is naturally aligned with the continuous parameter spaces of mineral processing setpoints. Future empirical work is strictly necessary to benchmark these algorithms against one another in simulated industrial environments to establish definitive operational hierarchies.

Translating this conceptual framework into operational reality requires overcoming severe computational bottlenecks inherent to industrial mining data. Unlike robotics simulators that permit millions of rapid evaluations, assessing the fitness and behavioral descriptors of a mining schedule or a blast design relies on computationally expensive discrete-event simulators or finite element analyses [38]. Consequently, the direct application of canonical MAP-Elites or CMA-ME is computationally intractable. Deployment in these high-dimensional spaces necessitates the integration of surrogate-assisted Quality–Diversity approaches, where computationally cheap regression models (e.g., Gaussian Processes or hierarchical neural networks [43]) approximate the objective and behavioral functions. Furthermore, mining environments are intrinsically stochastic due to geological uncertainty and sensor noise. Algorithms must incorporate uncertainty-handling mechanisms, such as adaptive sampling or noise-aware archive insertion rules [74,75], to ensure that the diverse operational strategies generated remain robust when transferred from simulation to the physical plant.

Following this framework, the implementation logic for three key areas is detailed below:

4.9.1. Predictive Maintenance and Asset Management

This can be formulated as generating a repertoire of maintenance policies. Analogous to robot damage recovery [14,16], mining needs policies to adapt to operational conditions.

Mathematical Formulation:

Let $\pi$ be a policy. Quality maximizes availability minus cost:

$$f\left(\pi \right)=\mathbb{E}\left[\sum _{t=0}^T{\gamma }^t\left(A_t\left(\pi \right)-\lambda C_t\left(\pi \right)\right)\right]$$

(20)

Descriptors map cost aggressiveness ($\overline{C}$) against risk profile (${\sigma }_{TBF}$):

$$\mathbf{b}\left(\pi \right)=\left(\begin{array}{c}\overline{C}\left(\pi \right)\\ {\sigma }_{TBF}\left(\pi \right)\end{array}\right)$$

(21)

A plant manager could select low-risk policies for critical bottlenecks and cost-optimized policies for auxiliary fleets, potentially learning from demonstrations [19,107].

4.9.2. Mineral Processing Optimization

The grinding-flotation circuit is a complex dynamic system [40,41,42]. Dynamic optimization [106] lacks diversity.

Mathematical Formulation: The Quality function maximizes the metallurgical recovery $R\left(\mathbf{u}\right)$, defined by the mass balance ratio between concentrate and feed:

$$f\left(\mathbf{u}\right)=R\left(\mathbf{u}\right)={\displaystyle \frac{M_{conc}\left(\mathbf{u}\right)·g_c\left(\mathbf{u}\right)}{M_{feed}·g_{feed}}}\times 100$$

(22)

where M denotes mass flow rates and g represents the ore grades. To ensure operational feasibility, the behavioral descriptors capture specific energy consumption (E), water usage (W), and the resulting concentrate grade ($g_c$):

$$\mathbf{b}\left(\mathbf{u}\right)=\left(\begin{array}{c}E\left(\mathbf{u}\right)\\ W\left(\mathbf{u}\right)\\ g_c\left(\mathbf{u}\right)\end{array}\right)$$

(23)

This allows adaptation to seasonal water restrictions [89].

4.9.3. Drilling and Blasting Planning

Blasting design involves trade-offs between cost, fragmentation, and environmental impact [38,39].

Mathematical Formulation: The Quality function aims to minimize the total unit cost, integrating drilling, explosive, and downstream processing costs:

$$f\left(\mathbf{d}\right)=C_{drill}\left(\mathbf{d}\right)+C_{blast}\left(\mathbf{d}\right)+C_{load}\left(\mathbf{d}\right)+C_{crush}\left(P_{80}\right)$$

(24)

where $C_{drill}$ and $C_{blast}$ represent direct operational costs, while $C_{load}$ and $C_{crush}$ account for the efficiency impact of fragmentation. The behavioral descriptors track the resulting fragmentation size ($P_{80}$) and vibration levels (PPV):

$$\mathbf{b}\left(\mathbf{d}\right)=\left(\begin{array}{c}{P}_{80}\left(\mathbf{d}\right)\\ \mathrm{PPV}\left(\mathbf{d}\right)\end{array}\right)$$

(25)

Engineers can select designs with low vibration near sensitive areas or fine fragmentation for crushing efficiency. Hierarchical surrogate models [43] could accelerate evaluation, while efficient species [31] and QD-RL [96] could scale these applications.

4.10. Threats to Validity and Limitations

It is important to acknowledge methodological limitations [79]. A potential threat to validity is the “Research Gap Artifact,” where the apparent lack of discrete QD studies could stem from the search strategy. However, the search string included specific terms like “discrete”, “combinatorial”, and “scheduling”, and the restriction to high-impact venues (Q1/Q2) ensures that if mature discrete QD methods existed, they would likely be captured. Additionally, potential publication bias and the language restriction to English apply. Finally, the mining application analysis is conceptual and theoretical; empirical validation with industrial partners remains a priority for future work.

Regarding the database selection strategy, the strict adherence to Web of Science and Scopus (filtering specifically for Q1/Q2 journals and top-tier CORE-ranked conferences) introduces a boundary effect on the perceived scarcity of discrete combinatorial studies. This high-impact inclusion criterion effectively captures mature, theoretically grounded algorithmic advancements—which are currently dominated by continuous robotics tasks and gradient-assisted methods. Consequently, early-stage applications of discrete QD, or domain-specific heuristic implementations published in non-indexed operational research workshops, specialized industry proceedings, or unverified preprints, may have been systematically excluded. Therefore, the identified gap should be interpreted precisely: it indicates a severe lack of consolidated, high-impact algorithmic formalization for discrete domains within the mainstream QD literature, rather than an absolute absence of preliminary exploratory attempts by isolated practitioners.

5. Conclusions

This systematic literature review synthesized the recent development of Quality–Diversity optimization as a distinct and rapidly maturing paradigm within evolutionary computation. The analysis shows that the field has expanded substantially in recent years, moving beyond canonical MAP-Elites formulations toward more scalable variants that incorporate covariance adaptation, gradient-informed search, and hardware acceleration.

At the same time, this review confirms that a major unresolved challenge remains in discrete combinatorial domains. While continuous-control applications dominate the current literature, the definition of robust diversity metrics and meaningful behavioral descriptors for non-continuous search spaces is still insufficiently consolidated. This gap is especially important because it limits the transfer of Quality–Diversity methods toward complex industrial problems in which solution diversity may have direct operational value.

From an applied perspective, this study highlights the strong potential of Quality–Diversity for resilient mining operations. By shifting the focus from a single optimal solution to a repertoire of diverse high-quality strategies, the paradigm offers a useful decision framework for environments characterized by uncertainty, operational variability, and competing performance criteria. In this context, predictive maintenance, mineral processing, and drilling and blasting emerge as promising directions for future implementation.

Overall, the main contribution of this review is to clarify the current state of the field, identify the discrete-domain methodological gap more precisely, and frame Quality–Diversity as a resilience-oriented optimization perspective with industrial relevance. Future research should therefore prioritize mathematically grounded diversity representations for combinatorial spaces, improved descriptor-learning strategies, and empirical validation in real-world industrial systems.Specifically, the development of standardized discrete combinatorial benchmark suites is urgently needed. Establishing these environments will enable rigorous head-to-head algorithmic comparisons of advanced QD variants, thereby grounding theoretical algorithmic recommendations in robust operational evidence and accelerating the transition of illumination algorithms into complex industrial settings.