Application of a Multi-Objective Optimization Algorithm Based on Differential Grouping to Financial Asset Allocation

Abstract

In the era of big data and rapid information growth, investors encounter a complex financial environment characterized by extensive data, conflicting investment objectives, and markets that are unpredictable due to economic and policy fluctuations. Hence, asset selection is vital for both investors and researchers. Multi-objective optimization algorithms balance multiple objectives to find optimal solutions and are widely used in engineering, economics, etc. This paper proposes a multi-objective decomposition optimization algorithm integrated with differential grouping (DG-MOEA/D). Initially, the algorithm employs the recursive spectral clustering differential grouping (RDGSC) technique to identify dependencies among variables, grouping them to reduce interactions between the variables. It then uses MOEA/D-UTEA to optimize each group, with an external archive for storing and updating solutions. Experimental results on the DTLZ and LSMOP test functions show that the DG-MOEA/D algorithm greatly outperforms the other seven comparison algorithms. When used in real-world scenarios like stock and bond asset allocation, the algorithm continues to outperform other methods, demonstrating its significant advantages in practical applications.

1. Introduction

The rapid progress of information technology has significantly transformed various industries, especially the financial sector. The growth of data sources, including high-frequency trading and macroeconomic indicators, has greatly increased the complexity of making investment decisions [1]. Investors must balance multiple, often conflicting objectives, such as maximizing returns, managing risks, and keeping liquidity. These challenges have reduced the effectiveness of traditional portfolio theories, like Markowitz’s framework [2], in today’s financial environment. As a result, developing better asset allocation methods has become a focus for both practitioners and academic researchers.

Multi-objective optimization problems (MOPs) constitute the foundational theoretical framework for addressing such conflicts. Their core involves transforming multiple interdependent objectives into a search for Pareto optimal solutions through mathematical modeling, rather than seeking the absolute optimal of a single objective [3]. Asset allocation inherently exhibits multi-objective characteristics: as investment scopes expand to encompass multiple asset classes such as stocks, bonds, and derivatives, the objective dimensions evolve from simple (return-risk) considerations to more complex dimensions, such as incorporating liquidity objective [4] or ESG metrics [5]. This results in a solution space characterized by complex traits, including high dimensionality, non-convexity, and discreteness. These properties diminish the efficacy of traditional optimization methods—such as weighted summation or ε-constraint approaches—because they depend on pre-selected parameter values or lack adequate local search capabilities.

The multi-objective evolutionary algorithm (MOEA), which uses swarm intelligence search and its ability for global convergence, has gradually become an effective tool for solving complex multi-objective optimization problems. Its main benefit is mimicking biological evolution—specifically selection, crossover, and mutation—to gradually find optimal solutions across multiple objectives via population evolution. This results in a well-distributed set of Pareto frontier solutions [6]. However, current MOEAs still encounter two main challenges when dealing with large-scale multi-objective optimization problems: variable coupling (non-linear dependencies among different asset price movements) and computational complexity (the solution space grows exponentially with the number of objectives) [7]. Although researchers have developed improved algorithms such as the decomposition-based MOEA/D [8] and the dominance-based NSGA-II [9], most techniques still rely on a ‘black-box’ optimization method. This method does not fully utilize the internal structural information among variables, which limits search efficiency and solution quality [10].

To address these limitations of current algorithms and enhance the efficacy of multi-objective optimization algorithms in financial asset allocation, this paper presents a differential grouping-based multi-objective optimization algorithm (DG-MOEA/D). The primary innovations of this algorithm are delineated as follows:

First, by analyzing the similarity between variables, it accurately identifies their inherent dependency relationships and carefully categorizes data into three types: completely separable, partially separable, and non-separable. This grouping method fully considers the data’s structure and characteristics, offering a more logical basis for subsequent optimization processes.

Secondly, for data that is completely separable, the optimization phase starts right away. The remaining data is further divided using an improved spectral clustering algorithm, which ensures high correlation within groups and low correlation between groups. This method effectively reduces data dimensionality, decreases computational complexity, and improves the efficiency and performance of the optimization algorithm.

Finally, the MOEA/D algorithm, combined with an external archive, is used to optimize each group separately. At the same time, the external archive mechanism stores and updates non-dominated solutions iteratively. This optimization approach enables effective trade-offs and coordination among multiple objectives, identifying superior solution sets to provide more scientifically sound and rational asset allocation proposals for investors.

To validate the effectiveness and superiority of the DG-MOEA/D algorithm, a series of experimental studies were conducted. Results on the DTLZ and LSMOP test functions show that the DG-MOEA/D algorithm significantly outperforms seven alternative algorithms: MOEA/D, MOEA/D-DU, SPEA2, NSGA-II, NSGA-III, IM-C-MOEA/D, and LERD. Additionally, when applied to practical scenarios such as asset allocation for a diversified set of instruments (including stocks, gold, and funds with varying liquidity profiles), the algorithm’s performance similarly exceeds other methods, clearly demonstrating its advantages in real-world applications.

The rest of this paper is organized as follows: Section 2 reviews related work on the algorithm, Section 3 explains the algorithm in detail, Section 4 provides experimental results and analysis, Section 5 applies the algorithm to financial asset allocation problems, and Section 6 summarizes the paper.

2. Related Work

2.1. Traditional Financial Asset Allocation Methods

Traditional portfolio theory and methods for equity investments have primarily depended on statistical models and empirical rules. Harry Markowitz [11] first proposed the mean-variance model for portfolio selection, presented as a quadratic programming formulation. Its objective is to minimize portfolio variance under given return constraints and full investment conditions. This pioneering work laid the foundations for modern portfolio theory, formalizing the risk-return problem as a single-objective optimization problem. Subsequently, researchers progressively transformed the Markowitz model into an equivalent two-objective optimization problem [12], simultaneously optimizing two conflicting objectives: maximizing expected returns while minimizing portfolio risk.

However, this model has notable flaws. It relies heavily on the strict market hypothesis, whose demanding assumptions often do not match real market conditions [13]. For instance, it assumes returns follow a normal distribution, whereas real-world markets may experience extreme events or exhibit fat-tailed distributions. Additionally, the model requires significant computational resources, and its use of covariance to measure risk is unclear and does not effectively show the exact size of potential losses investors could face [14]. As a result, many researchers have made improvements and developed new ideas based on the mean-variance model, suggesting different theories and methods.

In 1964, Sharpe developed the Capital Asset Pricing Model (CAPM) [15] based on portfolio and capital market theories. This model mainly explores the relationship between expected returns and risk in securities markets, serving as an equilibrium version of Markowitz’s mean-variance model under stronger market assumptions. However, the practical application of the standard CAPM is constrained by its reliance on assumptions that often fail to hold in dynamic real-world markets. While subsequent adjusted in Beta calculations (e.g., Vasicek [16] and option-adjusted [17]) have been proposed to address some of these limitations, the model in its basic form struggles to fully account for the evolving nature of security price volatility. Afterwards, portfolio theory experienced rapid growth to address these complexities.

The concept of Value-at-Risk (VaR) was first introduced in 1993. VaR represents the maximum potential loss an investment portfolio could face within a specified holding period at a given confidence level [18]. However, the VaR methodology has limitations in its foundational principles and statistical estimation methods. Focusing only on the objective probability of risk-related losses and emphasizing risk’s statistical features does not fully capture overall risk [19].

Addressing VaR’s limitations, Rockafeller and Uryasev [20] proposed a more effective risk metric: Conditional Value-at-Risk (CVaR). CVaR reflects the average excess loss an investor faces during a specified investment period, where losses above the VaR threshold are expected at a certain confidence level. Compared to VaR, CVaR considers the size of losses [21], producing results that more closely align with optimal values and providing a better measure of risks in real financial markets. As such, it has become a widely adopted measure for assessing tail risk.

Since Markowitz proposed the mean-variance theory, extensive literature has expanded or modified the basic model along three distinct pathways. The first approach focuses on simplifying the types and quantity of input data; the second concentrates on introducing alternative risk measurement methods; the third involves integrating additional criteria and/or constraints [22]. In this paper, we incorporate a liquidity objective to construct a three-objective portfolio model.

Liquidity refers to an asset’s capacity to be converted into cash swiftly without incurring significant value depreciation. Tobin defined liquid assets as those which can be realized immediately at market value [23]. In portfolio, liquidity not only determines an investor’s ability to respond promptly to redemptions, rebalancing, or unforeseen events, but also serves as a crucial bridge between risk control and return realization [24]. An asset that appears to offer superior returns may amplify losses during crises if illiquid, as timely disposal becomes impossible [25]. Consequently, liquidity serves as the third optimization objective in this portfolio model.

2.2. Multi-Objective Optimization Methods in Financial Asset Allocation

Multi-objective optimization algorithms provide significant benefits in real-world applications. They effectively balance conflicting objectives, choosing the best solutions based on specific situations and needs. As a result, they are widely applied across fields such as engineering, economics, logistics, and others, yielding notable results. For example, in engineering, they optimize product design parameters to find solutions that meet multiple objectives like performance, cost, and reliability. In economics, they enhance enterprise production planning by considering objectives such as efficiency, resource use, and cost control to develop the most rational strategies. In logistics, they optimize delivery routes to create the best distribution plans while addressing multiple aims, including delivery time, transport costs, and service quality.

Currently, multi-objective evolutionary algorithms (MOEAs) are broadly categorized into three types: dominance-based MOEAs, decomposition-based MOEAs, and indicator-based MOEAs. Some dominance-based MOEAs have been tested in benchmark experiments [26] or used in practical portfolio optimization problems [27,28]. For example, Mishra et al. conducted a comparative performance evaluation of three multi-objective methods—PAES, APAES, and NSGA-II—applying them to portfolio management [29]. Anagnostopoulos et al. applied NSGA-II, PESA, and SPEA2 algorithms to constrained two-objective portfolio optimization problems, showing that multi-objective evolutionary algorithms are generally effective and reliable strategies for solving these issues [30].

In recent years, variants of multi-objective swarm intelligence methods have also been applied to asset allocation problems. Examples include the non-dominated sorting multi-objective particle swarm optimization (NS-MOPSO) proposed by Mishra et al. to address portfolio selection problems involving constraints such as budget, lower bounds, upper bounds, and cardinality [31,32]; the multi-objective bacterial foraging optimization (MOBFO) applied to portfolio problems based on the Markowitz model [33]; KUMAR proposed a multi-objective co-variation based artificial bee colony (M-CABC), testing its efficacy on portfolio optimization benchmark problems from the OR-library [34]; Heung et al. introduced a novel dual-timescale composite neural-dynamical optimization method for quantity-constrained portfolio selection [35].

Nevertheless, existing multi-objective optimization algorithms exhibit numerous shortcomings and face significant challenges. For instance, computational complexity escalates dramatically when processing high-dimensional financial data due to excessive dimensionality, severely compromising algorithmic efficiency and performance. When addressing conflicting objectives, the absence of effective coordination mechanisms hinders the identification of solutions that achieve optimal trade-offs across multiple objectives. Furthermore, when applied to practical asset allocation problems, insufficient consideration of data dependencies results in suboptimal accuracy and reliability of optimization outcomes.

To address the above issues, this paper introduces the recursive spectral clustering differential grouping (RDGSC) technique and proposes the DG-MOEA/D algorithm. By integrating the decomposition-based multi-objective evolutionary algorithm (MOEA/D) with an external archiving mechanism, it fully leverages the strengths of each approach to enhance the algorithm’s performance in solving complex financial optimization problems. Specifically, DG-MOEA/D employs differential grouping to identify dependency structures among variables, enabling effective dimensionality reduction and grouped optimization for high-dimensional problems. Concurrently, it leverages the decomposition strategy and external archive mechanism of MOEA/D to enhance the algorithm’s search capability within the objective space and solution diversity. This approach better addresses challenges inherent in financial asset allocation, such as high dimensionality, conflicting objectives, and data dependencies. Compared to traditional multi-objective optimizers such as NS-MOPSO and M-CABC, DG-MOEA/D demonstrates superior comprehensive performance in convergence speed, solution quality, diversity preservation, and handling practical financial constraints, exhibiting greater applicability and practical value in financial contexts.

Decomposition-based multi-objective optimization algorithms [36,37,38] constitute a heuristic approach for addressing multi-objective optimization problems. This methodology first employs decomposition techniques to fragment large-scale global optimization problems into multiple smaller subproblems, subsequently resolving each subproblem through evolutionary algorithms. As subproblems typically exhibit reduced scale compared to the original problem, decomposition-based evolutionary algorithms can substantially diminish the required evaluation count via their decomposition mechanism.

Traditional decomposition methods are classified into dynamic and static approaches. Dynamic decomposition techniques involve the partitioning of variables throughout the evolutionary process, exemplified by methods such as Random Grouping (RG) [39]. This technique randomly allocates all variables into groups of equal size at each generation and then determines the group size based on historical fitness data. However, such methods fail to consider the fundamental structure of interactions among variables, which may result in related decision variables being separated into different groups and consequently leading to a significant decline in the performance of the algorithm.

Static grouping methods cluster decision variables during initialization and maintain these groups throughout the process. A D-dimensional problem is decomposed into k lower-dimensional subproblems [40], although this decomposition method ignores interactions among variables. The differential grouping method DG, proposed by Omidvar et al. [41], identifies interactions by monitoring fitness changes when variables are perturbed, grouping correlated variables together and significantly enhancing grouping accuracy. However, this method faces difficulties in detecting indirect interactions between variables. To address this challenge, Sun et al. proposed the extended differential grouping method XDG [42], which extends DG by considering both direct and indirect interactions concurrently. Mei et al. introduced an interaction structure matrix to formulate the global differential grouping method GDG [43]. This decomposition strategy may employ depth-first or breadth-first search to identify interactions between variables but requires extensive fitness evaluations (FEs) to detect such interactions. Sun et al. developed the recursive differential grouping method RDG [44], utilizing a recursive approach to detect interactions between variable sets, thereby significantly reducing FE usage. Lin et al. designed the DGSC method [45], which constructs an undirected weighted graph of decision variables and utilizes the design structure matrix, derived from differential values, as the similarity matrix for spectral clustering, resulting in more effective variable grouping. This paper applies the recursive spectral clustering differential grouping method to decrease FEs while enhancing data analysis accuracy and efficiency. A comparison of this grouping method with previously proposed approaches is presented in

Table 1. This table demonstrates the RDGSC grouping method in comparison with previously proposed grouping methods.

Grouping Method	XDG	RDG	RDGSC
Core Mechanism	Detects direct and indirect interactions	Recursively identifies nonlinear interactions between variables	Recursive spectral clustering with similarity matrix grouping
Accuracy Enhancement	High, though threshold-dependent	High, suitable for most CEC benchmarks	Superior, particularly in complex coupling and redundant variable scenarios
Feature Vector/Dimension Processing	No explicit dimensionality reduction	No explicit dimensionality reduction	Effectively reduces feature dimensions via spectral clustering
Suitable Problem Types	Moderate-scale problems with clear structure	Large-scale problems with separability	High-dimensional problems with complex coupling and nonlinearity

.

External archiving techniques function as an effective auxiliary method for multi-objective optimization, efficiently maintaining non-dominated solutions without significantly increasing the computational complexity of the algorithm. In recent years, numerous scholars have conducted comprehensive research on the application of external archiving mechanisms within evolutionary algorithms, notably improving the efficiency of population convergence. Michalak [46] integrated external populations into the evolutionary framework. By storing non-dominated solutions from each generation and selectively reintroducing some or all high-quality solutions into the primary population, this approach effectively enhanced the genetic diversity of the population. Improved algorithms utilizing external archives include PAES [47] and the JADE algorithm [48], both featuring adaptive external archives. These methods preserve all non-dominated solutions identified during evolution, thereby effectively preventing the loss of high-quality solutions due to random factors. During multi-objective optimization, this paper employs the MOEA/D-UTEA algorithm, which adopts the MOEA/D framework while incorporating an external archive mechanism. This approach retains superior genetic traits within the parent generation, preventing the loss of high-quality individuals during the search process and consequently further improving the algorithm’s optimization performance.

3. The Framework of MOEA/D Combined with Differential Grouping

3.1. General Framework

This paper introduces the DG-MOEA/D algorithm, which integrates differential grouping, the MOEA/D algorithm, and an external archive. The algorithm initially employs a recursive differential grouping method to precisely identify data that is completely separable, partially separable, or non-separable. Subsequently, it utilizes an enhanced spectral clustering differential grouping approach to partially separable or non-separable data, leveraging spectral clustering’s capacity to handle complex similarity relationships to enhance grouping accuracy.

During the multi-objective optimization stage, the algorithm is rooted in the MOEA/D framework and incorporates an external archive mechanism. By decomposing the multi-objective problem into multiple single-objective subproblems via non-dominated sorting and maintaining a set of high-quality solutions through the external archive, the algorithm directs the population towards the Pareto front.

Figure 1 shows the overall framework diagram of the DG-MOEA/D algorithm. Algorithm 1 outlines the framework of DG-MOEA/D, with details of RDGSC provided in Algorithm 2 and details of MOEA/D-UTEA in Algorithm 5.

Algorithm 1: DG-MOEA/D Algorithm

Input: $N$(Population size) $T$ (Neighborhood size) $k$ (Group number) $D$ (Problem dimension) Output: $F$ (Final population) $1 \mathrm{Groups} \leftarrow$$\mathrm{RDGSC}\left(D,k\right)$ 2 $\mathbf{for} \mathrm{each} G_i\in$ Groups do $3\hspace{1em} \mathrm{Generate} \mathrm{weight} \mathrm{vectors} \{{\lambda }^1,\cdots ,{\lambda }^N\}$$\mathrm{for} \mathrm{group} G_i$; $4\hspace{1em} \mathrm{Compute} \mathrm{neighborhood} B\left(i\right)$$\mathrm{for} \mathrm{each} \mathsf{\lambda }$; $5\hspace{1em} \mathrm{P} \leftarrow$$\mathrm{InitializePopulation}(N$); $6\hspace{1em} z^{\ast }\leftarrow$ InitializeReferencePoint( ); $7\hspace{1em} F_i\leftarrow$$\mathrm{MOEA}/\mathrm{D}-\mathrm{UTEA}\left(G_i,N,T\right)$; $8\hspace{1em} F\leftarrow F\cup F_i$ 9 end for

3.2. Recursive Spectral Clustering Differential Grouping

In large-scale optimization problems (LSGOs), each decision variable affects the variation of the objective function value. An LSGO problem can be defined as follows:

$$\min f\left(\overrightarrow{X}\right),\overrightarrow{X}=\{x_1,x_2,\cdots ,x_n\}\in {\mathbb{R}}^n$$

(1)

where $f\left(\overrightarrow{X}\right)$ denotes the objective function, $\overrightarrow{X}$ is the decision vector of $f\left(\overrightarrow{X}\right)$, and for each dimension of $\overrightarrow{X}$, the decision variables are ${\{x}_1,x_2,\cdots ,x_n\}$, where $n$ denotes the dimension of the function $f\left(\overrightarrow{X}\right)$.

If the decision variables influence one another, their effects on the objective function value changes are not mutually independent. Variable interactions may be categorized into three scenarios: direct interaction, indirect interaction, and mutual independence, defined as follows:

Definition 1.

Let f be a differentiable function. If a candidate solution  ${\overrightarrow{x}}^{\ast }$ exists, and the following condition holds:${ \partial }^{2}f\left({\overrightarrow{x}}^{\ast }\right)/\partial x_i\partial x_j\ne 0$ , then decision variables$x_i$ and$x_j$ interact directly, denoted by$x_i\leftrightarrow x_j$ . Otherwise,$x_i$ and$x_j$ interact indirectly or are independent.

Definition 2.

Let f be a differentiable function. For any candidate solution  ${\overrightarrow{x}}^{\ast }$ , if the following conditions hold:${ \partial }^{2}f\left({\overrightarrow{x}}^{\ast }\right)/\partial x_i\partial x_j=0$ , and a set of decision variables$\{x_{k1}, x_{k2},\cdots , x_{kt} \}\subset \overrightarrow{X}$ exists, such that$x_i\leftrightarrow x_{k1}\leftrightarrow x_{k2}\leftrightarrow \cdots \leftrightarrow x_{kt} \leftrightarrow x_j$ , decision variables$x_i$ and$x_j$ interact indirectly.

Definition 3.

If neither of the above conditions for direct or indirect interaction is satisfied, then$x_i$and$x_j$are said to be mutually independent.

To verify the interactions between variables, Omidvar et al. proposed the following criterion: if

$$|{\mathsf{\Delta }}_1-{\mathsf{\Delta }}_2|>\epsilon$$

(2)

then variables $x_i$ and $x_j$ are non-separable (i.e., interaction exists). Here ${\mathsf{\Delta }}_1=f\left(\cdots ,{\overline{x}}_i,\cdots ,x_j,\cdots \right)-f\left(\cdots ,x_i,\cdots ,x_j,\cdots \right)$,${\mathsf{\Delta }}_2=f\left(\cdots ,{\overline{x}}_i,\cdots ,{\overline{x}}_j,\cdots \right)-f\left(\cdots ,x_i,\cdots ,{\overline{x}}_j,\cdots \right)$, and ${\overline{x}}_i=x_i+\mathsf{\Delta }{x}_i$, ${\overline{x}}_j=x_j+\mathsf{\Delta }{x}_j$ represent perturbations on variables $x_i$ and $x_j$ respectively. The threshold $\epsilon$ is introduced to avoid errors due to computational rounding. If Equation (2) does not hold, the decision variable $x_i$ is considered a separable variable.

Although differential grouping effectively identifies interactions among variables, its computational complexity remains high, rendering it impractical for ultra-high-dimensional problems. To address this challenge, Sun et al. introduced Recursive Differential Grouping (RDG) [44], which utilizes a binary search strategy to reduce computational complexity and significantly improve decomposition efficiency. RDG no longer detects interactions pairwise; instead, it identifies interactions between sets of variables, thereby facilitating effective decomposition for large-scale problems. While RDG and its variants demonstrate considerable efficiency advantages, their accuracy in decomposition can still be improved, particularly when dealing with overlapping interactions or highly complex subcomponents. Spectral clustering is a graph-based clustering method that treats data points as nodes within a graph. It constructs a graph structure based on similarities (or connectivity) between nodes, then utilizes the spectral properties of the graph—namely the eigenvalues and eigenvectors of the Laplacian matrix—to achieve clustering. The fundamental principle of spectral clustering is to partition the graph’s nodes into several clusters such that connections within clusters are strong while connections between clusters are weak. Inspired by clustering principles—where objects within the same cluster exhibit high similarity while those across different clusters show stark differences—we combine differential grouping with spectral clustering to propose the Recursive Spectral Clustering Differential Grouping (RDGSC) algorithm, as detailed in Algorithm 2. This algorithm seeks to maximize interactions within subcomponents while minimizing interactions between them. The $\left|\Delta f\left(x_i,x_j\right)\right|$ mentioned in the algorithm measures the strength of interaction between variables, corresponding to $\left|{\mathsf{\Delta }}_1-{\mathsf{\Delta }}_2\right|$ in Equation (2). It is computed by perturbing variables and observing changes in the objective function value.

Algorithm 2: RDGSC Algorithm

Input:$f,ub,lb,vars,D,k,\epsilon$ Output: Groups $1 \mathbf{for} \mathrm{each} v$ in vars do $2\hspace{1em} \mathbf{if} \forall j\ne i,\left|\Delta f\left(x_i,x_j\right)\right|<\epsilon$ then $3\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Groups}[\mathrm{fully}\_\mathrm{separable}] \leftarrow v \cup$ Groups[fully_separable] 4    remaining_vars ← vars- Groups[fully_separable] 5 end for $6 \mathbf{if} \mathrm{remaining}\_\mathrm{vars} \ne \mathrm{\varnothing }$ then $7\hspace{1em} \mathsf{\Theta } \leftarrow \mathrm{BuildSimilarityMatrix} (f$$, \mathrm{remaining}\_\mathrm{vars}, lb$$, ub$); $8\hspace{1em} \mathrm{clustered}\_\mathrm{vars} \leftarrow \mathrm{DGSC} (\mathsf{\Theta }, \min (k$, len (remaining_vars)));

The DGSC algorithm requires the construction of a similarity matrix to provide input for subsequent spectral clustering. This achieves rational variable grouping by quantifying the interaction strength between decision variables. Initially, the algorithm detects whether an interaction exists between each pair of variables based on the principle of differential grouping. Specifically, for any two variables $x_i$ and $x_j$, it applies minor perturbations to each and observes changes in the objective function value to determine whether they satisfy the interaction condition. If Equation (2) holds, these variables are deemed to interact directly. Subsequently, the algorithm organizes the interaction detection results for all variable pairs into a design structure matrix (DSM) $\mathsf{\Theta }$. The elements ${\mathsf{\Theta }}_{ij}$ of this matrix are defined as follows: if variables x and x exhibit interaction (including direct or indirect interaction), then ${\mathsf{\Theta }}_{ij}$= 1; otherwise, ${\mathsf{\Theta }}_{ij}$= 0. This matrix is symmetric, i.e., ${\mathsf{\Theta }}_{ij}$=${\mathsf{\Theta }}_{ji}$, and the diagonal elements are typically set to 1 or 0 depending on whether the variable’s interaction with itself is considered.

Upon completion of the DSM matrix, it is directly employed as the similarity matrix $\mathrm{W}$, i.e., $\mathrm{W}=\mathsf{\Theta }$. This signifies that similarity between variables is determined solely by whether interactions exist between them: interactions yield a similarity value of 1, while the absence of interactions results in a value of 0. This construction method is straightforward and succinct, effectively reflecting the structural relationships among variables.

Subsequently, the algorithm proceeds to the spectral clustering stage: first, the diagonal degree matrix $\mathrm{D}$ is computed from the similarity matrix $\mathrm{W}$, where each diagonal element $\mathrm{D}(i,i)={\sum }_{j=1}^n{w}_{ij}$ represents the total similarity (i.e., connectivity) between variable $x_i$ and all other variables; Next, the unnormalized Laplacian matrix $\mathrm{L}=\mathrm{D}-\mathrm{W}$ is constructed; then, using MATLAB’s (2022) eigs function, $\mathrm{L}$ undergoes eigenvalue decomposition to extract the eigenvectors corresponding to the first k smallest eigenvalues, forming a new feature space Q. Within this space, structurally similar variables cluster closer together; Finally, perform K-means clustering on variables within the feature space Q. This divides the original variables into $k$ groups, achieving strong interactions within groups and weak interactions between groups. This optimises grouping and outputs the variable grouping result ‘Groups’ for subsequent optimization. The detailed process is outlined in Algorithm 3.

Algorithm 3: DGSC Algorithm

Input:$\mathsf{\Theta }, k$ Output: Groups $1 \mathrm{W} \leftarrow$ Θ; $2 \mathrm{D} \leftarrow 0_{n\times n}$; $3 \mathbf{for} i$$= 1 \to$$n$ do $4 \mathrm{D}(i$$,i$$) ={\sum }_{j=1}^n{w}_{ij}$ 5 end for 6 L = D − W; $7 (\mathrm{Q},\mathrm{V}) = \mathrm{eigs}(\mathrm{L}, k$, ’SA’); $8 \mathrm{C} = \mathrm{K}-\mathrm{means}(\mathrm{Q}, k$); $9 \mathrm{Groups} = \mathrm{cell}\left(1,k\right)$; $10 \mathbf{for} i=1\to k$ $11 \mathrm{Groups}\left[i\right]=\mathrm{find}\left(C==i\right)$ 12 end for

3.3. MOEA/D Algorithm Combined with External Archives

MOEA/D achieves a Pareto front by decomposing a multi-objective optimization problem into several single objective optimization problems and optimizing them simultaneously. One of the most frequently used decomposition methods is Tchebycheff approach. This approach transfers vector optimization into scalar optimization with a weight vector λ and a reference point $z^{\ast }$ as follows (i.e., minimization example), where $z_m^{\ast }$ is the minimum value of $m$-th objective. By using this method, the retrieved optimal is the intersection of Pareto Front and the straight line determined by $\lambda$ and $z^{\ast }$.

$$\min g^{te}\left(x∣\lambda ,z^{\mathrm{*}}\right)=\underset{m=1,\cdots ,M}{\max }\left\{{\lambda }_m\left|f_m\left(x\right)-z_m^{\mathrm{*}}\right|\right\}$$

(3)

$$\mathsf{\lambda }=\left({\mathsf{\lambda }}_1,\cdots ,{\mathsf{\lambda }}_M\right),{\mathsf{\lambda }}_1+\cdots +{\mathsf{\lambda }}_M=1$$

(4)

$$z^{\mathrm{*}}=\left(\min f_1,\cdots ,\min f_M\right)$$

(5)

In MOEA/D, everyone in the population is optimized as a solution to a single-objective optimization problem, which is decomposed using a weight vector. During the optimization process, the solution utilizes information from its neighbors. Neighboring individuals refer to solutions to neighboring single-objective optimization problems, which can be calculated using the shortest Euclidean distance between weight vectors. When reproducing offspring, MOEA/D selects parents from the neighbors of the current individual. After offspring are generated, all neighbors can be used to update the current individual.

In MOEA/D, each individual in the population is optimized as a solution to a single-objective subproblem, decomposed from the original multi-objective problem using a weight vector ${\lambda }^i$. The algorithm leverages neighborhood collaboration to enhance efficiency: the neighborhood B($i$) of the $i$-th subproblem consists of indices of the T closest weight vectors to ${\lambda }^i$, determined by minimizing the Euclidean distance $d\left({\lambda }^i,{\lambda }^j\right)=\sqrt{{\sum }_{k=1}^m{\left({\lambda }_k^i-{\lambda }_k^j\right)}^2}$,where $m$ is the number of objective functions. During optimization, parents for reproduction are selected from B($i$), ensuring offspring inherit traits from similar subproblems. New solutions then update all neighbors in B($i$) if they improve the corresponding subproblems. This localized approach balances exploration and exploitation by restricting information exchange to structurally related solutions, thereby maintaining diversity while accelerating convergence.

This paper introduces an external archive mechanism as shown in Algorithm 4 and combines it with the MOEA/D algorithm. The MOEA/D-UTEA algorithm process is shown in Algorithm 5.

Algorithm 4: External Archive Mechanism

Input: $\hspace{1em}F$ (Selected set of non-dominated layer solutions); $\hspace{1em}{F}_k$ (The last nondominated layer of the Nth selected solution); $\hspace{1em}N$ (Maximum archive capacity) Output: ExArchCollection (Updated collection of external archives) $1 \mathbf{if} |F$$| == N$ then $2\hspace{1em} \mathrm{Add} F$ to ExArchCollection; $3 \mathbf{if} |F$$|! = N$ then $4\hspace{1em} \mathbf{for} X_i\in F_k$ do $5\hspace{1em}\hspace{1em} \mathrm{Calculate} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{solution} X_i$$\mathrm{and} \mathrm{all} \mathrm{weight} \mathrm{vectors}, \mathrm{find} \mathrm{the} \mathrm{nearest} \mathrm{weight} {\mathsf{\lambda }}_j$; $6\hspace{1em}\hspace{1em} \mathbf{if} \mathrm{solution} X_i$$\mathrm{cannot} \mathrm{be} \mathrm{dominated} \mathrm{by} \mathrm{any} \mathrm{solution} \mathrm{in} {\mathsf{\lambda }}_j$ then $7\hspace{1em}\hspace{1em} \mathrm{Add} \mathrm{solution} X_i$ to ExArchCollection; $8\hspace{1em}\hspace{1em} \mathbf{if} \mathrm{solution} X_i$$\mathrm{dominates} \mathrm{any} \mathrm{solution} \mathrm{in} {\lambda }_j$ then $9\hspace{1em}\hspace{1em} \mathrm{The} \mathrm{dominated} \mathrm{solutions} \mathrm{in} {\mathsf{\lambda }}_j$ are deleted; $10\hspace{1em}\hspace{1em} \mathbf{if} \mathrm{solution} X_i$$\mathrm{and} \mathrm{solutions} \mathrm{in} {\mathsf{\lambda }}_j$ are non-dominated then $11\hspace{1em}\hspace{1em} \mathrm{Preserve} \mathrm{solutions} \mathrm{closer} \mathrm{to} \mathrm{the} \mathrm{weight} \mathrm{vector} {\mathsf{\lambda }}_j$; 12  end for $13 \mathbf{while} | \mathrm{ExArchCollection} | > N$ then $14\hspace{1em} \mathbf{for} F\left(x_i\right)\in$ ExArchCollection do $15\hspace{1em}\hspace{1em} \mathbf{for} F\left(x_i\right)\in F_{k-1}$ do $16\hspace{1em}\hspace{1em} \mathrm{angle}\_\mathrm{matrix} \leftarrow$ Calculate the angle between each pair of solutions; 17   end for 18   Delete the solution with minimum angle; 19  end for 20 end while

When the algorithm is initiated, the external archive is initialized as the set of non-dominated solutions in the population. A non-dominated solution refers to a solution in a multi-objective optimization problem where no other solution can outperform it across all objectives. The objective function values of the population are sorted using non-domination sorting to identify all individuals in the first layer (i.e., non-dominated solutions). After each iteration, the external archive is updated by first merging the current external archive with the new population into a single set, then using the function to identify the non-dominated solutions within the merged set, and finally reassigning these non-dominated solutions to the EA. This ensures that the external archive always retains all non-dominated solutions found up to the current iteration, representing the optimal solution set for the multi-objective optimization problem.

Algorithm 5: MOEA/D-UTEA Algorithm

Input: $\hspace{1em}T$ (Neighborhood size); $\hspace{1em}N$ (Population size); max_gen (Maximum number of iterations) Output: $\hspace{1em} F$ (Final population) $1 \mathrm{Groups} \leftarrow$$\mathrm{RDGSC}(\mathrm{D},k$); $2 \mathrm{Generate} \mathrm{uniform} \mathrm{weight} \mathrm{vectors} {\mathsf{\lambda }}^1,\cdots ,{\mathsf{\lambda }}^N$; $3 \mathrm{Compute} \mathrm{neighborhood} B\left(i\right)$$\mathrm{for} \mathrm{each} {\mathsf{\lambda }}^i\left(i=1,\cdots ,N\right)$ $4 \mathrm{P} \leftarrow$$\mathrm{InitializePopulation}(N$); $5 \mathrm{z}* \leftarrow$ InitializeReferencePoint( ); $6 \mathrm{ExArch} \leftarrow \mathrm{\varnothing }$; 7 While gen < max_gen do $8\hspace{1em}\hspace{1em} \mathbf{for} \mathrm{each} x^i$ do $9\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{parents} \leftarrow$$\mathrm{SelectFromNeighborhood}(\mathrm{P}, \mathrm{B}(x^i$)); $10\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{offspring} \mathrm{y} \leftarrow$ RecombinationAndMutation(parents); $11\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{z}* \leftarrow$ UpdateReferencePoint(z*, y); $12\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{for} \mathrm{each} x^p$$\mathrm{in} \mathrm{neighbor} \mathrm{of} x^i$ do $13\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathbf{if} g^{te}\left(y∣{\mathsf{\lambda }}^p,z^{\mathrm{*}}\right)\le g^{te}\left(x^p∣{\mathsf{\lambda }}^p,z^{\mathrm{*}}\right)$ then $14\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x^p\leftarrow y$ 15   end for 16    end for 17 end while

After updating the external archive, call the function to perform a truncation operation on it to maintain the diversity of solutions. The function first calculates the cosine angle between all solutions in the external archive, using the MATLAB (2022) function pdist2 to calculate the distance between each pair of vectors in the two matrices, with cosine as the distance metric, i.e., calculating the cosine distance. The formula for the cosine distance is as follows: For two vectors $a$ and $b$, their cosine distance $d_{cosine}=1-{\displaystyle \frac{a\cdot b}{\left|a\right|\left|b\right|}}$, where $a\cdot b$ is the dot product of the vectors, and $\left|\right|a\left|\right|$ and $\left|\right|b\left|\right|$ are the norms of the vectors. The function sorts the solutions based on the angle between them, retaining solutions with larger angles in the evolutionary algorithm and removing those with smaller angles. This prevents overly similar solutions from existing in the EA, ensuring a more uniform distribution of solutions in the target space and thereby improving the algorithm’s search performance.

3.4. Computational Complexity Analysis

The computational complexity of DG-MOEA/D is analyzed as follows: recursive spectral clustering differential grouping has a computational complexity of $O\left(n^3\right)$, Laplace matrix calculation has a high computational complexity, and the time complexity of constructing similarity matrices, feature decomposition, and similarity verification are all $O\left(n^2\right)$. The time complexity of the MOEA/D-UTEA algorithm is $O(G·m·N^2)$, with the main computational overhead arising from the maintenance of the external archive, including non-dominated sorting and angle screening. Here, $n$ represents the number of samples, G denotes the number of iterations, N is the population size, and $m$ is the number of objectives.

The space complexity is $O\left(n^2+N·\left(d+m\right)\right)$, dominated by the similarity matrix and optimization storage. If the data scale is large, where “large” in contemporary scientific computing is not absolute, typically referring to data approaching or exceeding single-machine memory capacity, requiring extremely long processing times (days to weeks), or necessitating multi-machine parallel processing (such as GPU clusters or distributed computing), then in spectral clustering, similarity calculations and feature decomposition may become bottlenecks, as these operations significantly increase computational complexity as the data size grows; meanwhile, the time-consuming operations in the optimization phase are non-dominated sorting and archive maintenance.

3.5. Three-Objective Portfolio Model

The goal of financial asset allocation is to allocate a finite set of assets in a reasonable manner to better balance returns and risks. Markowitz’s seminal work [11] initially modelled portfolio optimization as a single-objective problem, typically either maximizing return for a given level of risk or minimizing risk for a given level of return. This uni-objective Markowitz model can be equivalently expressed as a bi-objective optimization problem for the purpose of Pareto analysis, where the two conflicting objectives are to maximize return and minimize risk, as shown in Formula (6).

$$\begin{gathered} \underset{w}{\max } RETURN={\sum }_{i=1}^n{r}_i{w}_i \\ \underset{w}{\min } RISK={\sum }_{i=1}^n{\sum }_{j=1}^n{\mathsf{\sigma }}_{ij}{w}_i{w}_j \\ \mathrm{subject}\mathrm{to},{\sum }_{i=1}^n{w}_i=1,\left(0\le w_i\le 1\right) \end{gathered}$$

(6)

where $w=\left(w_1,w_2,\cdots ,w_n\right)$ is a vector representing the investment ratios of $n$ assets, $r_i$ is the rate of return of the $i$-th asset, and ${\mathsf{\sigma }}_{ij}$ is the covariance between the rates of return of the $i$-th asset and the $j$-th asset.

However, in the complex reality of the stock investment market, investors must optimize their decisions within a three-dimensional framework of return, risk, and liquidity. Specifically, in addition to the traditional objectives of maximizing expected returns and minimizing risk metrics (CVaR), asset liquidity—a critical factor influencing the operability of a portfolio—must also be incorporated into the optimization framework. Based on this, this paper constructs a three-objective portfolio optimization model that accounts for transaction cost constraints. The core objective function is defined as follows: (1) Maximize expected returns; (2) Minimize conditional value at risk (CVaR); (3) Maximize the comprehensive liquidity indicator.

Within the three-objective model, complex interactions exist between CVaR, returns, and liquidity. For instance, certain high-yield assets may simultaneously exhibit elevated CVaR and low liquidity, necessitating investors to perform nuanced trade-offs between these three factors when constructing portfolios. Introducing CVaR as a risk metric assists investors in conducting a more comprehensive assessment of portfolio risk, particularly during market extremes. Simultaneously, by considering liquidity factors, investors can avoid the inability to adjust portfolios promptly due to insufficient asset liquidity, thereby mitigating potential loss risks. This interplay can be represented through a three-objective optimization model. Utilizing a multi-objective optimization framework enables the coordinated allocation of portfolio profitability, security, and liquidity, identifying a portfolio configuration that performs well across diverse market conditions. Its specific mathematical expression is as follows:

Firstly, the formula for maximizing expected returns is:

$${\displaystyle \underset{w}{\max }} E\left(\omega \right)=\sum _{i=1}^n{r}_i{w}_i-\sum _{i=1}^{n}c{w}_i$$

(7)

where $\omega$ is the asset decision vector, $r_i$ is the expected return on asset $i$, c is the transaction cost, and $w_i$ is the weight of asset $i$ in the investment portfolio. It should be noted that this paper simplifies transaction costs into a linear function, including core elements such as bid-ask spread predictions, brokerage fees, and exchange stamp duties, primarily to serve the needs of quantitative analysis within a multi-objective optimization framework. In real-world scenarios, transaction costs are more complex, potentially involving dynamic factors such as slippage costs, market impact costs, and implicit settlement fees in addition to the above costs.

Secondly, following its theoretical advantages, CVaR is adopted as the risk metric to be minimized in our model. The traditional Markowitz framework relies on the mean and covariance of historical data to estimate future returns and risks, but this method often exhibits suboptimal performance on out-of-sample data and is vulnerable to the “error maximization” effect [49]. In contrast, CVaR not only accounts for average losses but also specifically focuses on extreme losses exceeding the Value at Risk threshold. This allows our model to more comprehensively capture the potential risks of a portfolio under extreme market conditions. By emphasizing worst-case scenarios, the use of CVaR enhances the robustness of the optimization results. Its calculation formula is shown in Equation (8). However, it should be noted that if only market risk is considered while neglecting the investor’s personalized risk tolerance, the practical utility of CVaR-based models will be limited [50].

$$\underset{w,\mathsf{\alpha }}{\min } F\left(\omega ,\mathsf{\alpha }\right)=\mathsf{\alpha }+{\displaystyle \frac{1}{T\left(1-\mathsf{\beta }\right)}}\sum _{t=1}^T{\left[\mathsf{\psi }\left(\omega ,y_t\right)-\mathsf{\alpha }\right]}^{+}$$

(8)

where $\mathsf{\omega }$ is the asset decision vector, $\mathsf{\alpha }$ is the risk level parameter, $T$ is the number of time periods, $\mathsf{\beta }$ is the confidence level parameter, $\mathsf{\psi }\left(\mathsf{\omega },y_t\right)={\sum }_{i=1}^n\left(-y_{it}{\mathsf{\omega }}_i\right)$ is the expected loss of the asset portfolio under specific weights, and $y_{it}$ is the return of asset $i$ in period $t$.

Thirdly, the liquidity indicator $L\left(\omega \right)$ is used to measure the liquidity of assets in an investment portfolio. In this paper, we represent portfolio liquidity as a linearly weighted form as shown in Equation (9).

$$\underset{w}{\max } L\left(\mathsf{\omega }\right)=\sum _{i=1}^n{w}_i\cdot l_i$$

(9)

where $w_i$ is the weight of asset $i$, $l_i$ is the liquidity indicator of asset $i$. To quantify $l_i$ concretely, this paper employs the reciprocal of Amihud’s illiquidity metric to measure asset liquidity. The Amihud indicator reflects price volatility per unit transaction value of an asset. A higher illiquidity ratio ($ILLI{Q}_i$) indicates poorer liquidity. Therefore, to construct a metric $l_i$ positively correlated with liquidity, we take its reciprocal:

$$l_i=1/ILLI{Q}_i=N/{\sum }_{d=1}^N{\displaystyle \frac{\left|R_{i,d}\right|}{Q_{i,d}}}$$

(10)

where $N$ denotes the total number of trading days within the calculation period, $R_{i,d}$ represents the return on asset $i$ on the $d-$th trading day, and $Q_{i,d}$ denotes the trading volume of asset $i$ on the $d-$th trading day.

In addition, the sum of the weights of all assets in the investment portfolio must equal 1, i.e., ${\sum }_{i=1}^n{w}_i=1$, and $0\le w_i\le 1,i=\mathrm{1,2},\cdots ,n.$

Ensure that the weight distribution of the investment portfolio is reasonable, with no asset weights being negative or exceeding their appropriate proportions. Asset weights $w_i\ge 0$ indicate that short selling of any asset is not permitted in the investment portfolio. Short selling refers to investors borrowing assets and immediately selling them, aiming to repurchase them at a lower price in the future and return them. This practice is prohibited to prevent an escalation in the risk associated with the investment portfolio.

4. Experiments and Analysis

4.1. Test Functions

To assess the feasibility of the DG-MOEA/D algorithm, this paper selected the DTLZ and LSMOP series of functions as test problems. The selected test functions encompass various types of optimization challenges, including linear, nonlinear, convex, and concave problems, thereby enabling a comprehensive evaluation of the algorithm’s performance across diverse problem categories and ensuring its broad applicability and robustness. Furthermore, DTLZ functions can be readily extended to arbitrary numbers of objectives and decision variables, allowing for flexible adaptation of the objectives’ quantity and the decision variables’ dimensions in accordance with specific experimental requirements. The LSMOP functions, characterized by their large number of decision variables, exhibit high-dimensional features, with more complex data distributions and intrinsic properties, necessitating that the algorithm address issues such as the curse of dimensionality. Consequently, LSMOP functions can effectively evaluate the algorithm’s search capability and convergence speed in high-dimensional spaces.

4.2. Evaluation Metrics

In the performance evaluation of multi-objective optimization algorithms, inverse generation distance (IGD) and hypervolume (HV) are two important metrics. This paper uses these two metrics to measure their overall performance. The smaller the IGD value, the closer the Pareto Front (PF) formed by the final solution set obtained by the algorithm to the true PF; the larger the HV value, the better the overall performance of the algorithm. Below, we will introduce these two metrics in detail. IGD represents the mean distance between the actual Pareto optimal solution set, denoted as $P^{\ast }$, and the solution set $P$ generated by the algorithm, assuming a uniform distribution across the Pareto front. It is formally defined as follows:

$$IGD={\displaystyle \frac{1}{\left|P^{\mathrm{*}}\right|}}\sum _{v\in P^{\mathrm{*}}}d\left(v,P\right)$$

(11)

where $d(v,P)$ denotes the minimum Euclidean distance between individuals in population $P$ and individual $v$; $P^{\ast }$ denotes a certain number of individuals uniformly selected from the true PF; and $P$ denotes the optimal solution set obtained by the algorithm. IGD serves as an evaluation metric for the overall performance of the algorithm, reflecting its distributional and convergent properties, with smaller values being more desirable. When the IGD value is very small, it indicates that the optimal solution set obtained by the algorithm performs well in terms of distributivity and convergence.

The hypervolume metric is employed to quantify the set of non-dominated solutions produced by multi-objective optimization algorithms, as well as the volume of the hypercube within the objective space bounded by reference points. The mathematical expression is illustrated below:

$$HV\left(S,Z^{\mathrm{*}}\right)=\mathsf{\delta }\left(\bigcup _{i=1}^{\left|S\right|}{v}_i\right)$$

(12)

where $\mathsf{\delta }$ represents the Lebesgue measure, which is used to measure volume; $\left|S\right|$ denotes the number of non-dominated solutions in the solution set; and $v_i$ denotes the hypercube formed by the reference point $Z^{\mathrm{*}}$, and the $i$-th solution in the solution set. HV is an effective univariate quality metric that exhibits strict monotonicity regarding Pareto dominance. The larger the HV value, the better the performance of the corresponding algorithm.

4.3. Comparison Algorithms

To verify the performance of the proposed DG-MOEA/D algorithm, six classical and recently introduced MOEA algorithms were employed for comparison purposes. The following provides a concise description of each comparative algorithm:

• Multi-Objective Evolutionary Algorithm Based on Decomposition (MOEA/D): Utilizes weight vectors or decomposition strategies to convert complex multi-objective problems into multiple single-objective sub-problems, and shares information through a neighborhood collaboration mechanism, thereby improving search efficiency while ensuring uniformity in the solution set distribution.
• Nondominated Sorting Genetic Algorithm II (NSGA-II): Achieves efficient multi-objective optimization through rapid nondominated sorting and crowding distance calculation, while utilizing an elite retention strategy to ensure solution diversity and convergence.
• Nondominated Sorting Genetic Algorithm III (NSGA-III): Proposed for high-dimensional multi-objective optimization problems, it introduces a reference point mechanism to guide population evolution, combining nondominated sorting with adaptive normalization strategies to achieve uniform solution distribution while maintaining solution set diversity.
• Strength Pareto Evolutionary Algorithm 2 (SPEA2): By introducing a fine-grained fitness allocation strategy, density estimation techniques, and an enhanced archive truncation method, it enhances population diversity while preserving Pareto solutions, thereby avoiding premature convergence.
• MOEA/D with a Distance-Based Updating Strategy (MOEA/D-DU): Updates individuals in the population using distance metrics. This strategy randomly selects a solution in each iteration and updates it based on its distance from other solutions in the population.
• An Inverse Modelling Constrained MOEA/D (IM-C-MOEA/D): A multi-objective evolutionary algorithm combining decomposition strategies with constraint handling mechanisms, it uses inverse modelling techniques to map the objective space to the decision space, effectively addressing constrained real-world optimization problems.
• Large-scale Multi-objective Optimization via Reformulated Decision Variable Analysis (LERD): This algorithm reformulates the decision variable analysis process into an optimization problem with binary decision variables, enabling efficient large-scale multi-objective optimization, significantly reducing computational complexity, and improving optimization efficiency.

Each comparison method was run independently 30 times on each test problem, and the Wilcoxon rank-sum test was used to compare the results at a significance level of 0.05. The symbols ‘+’, ‘−’, and ‘=’ in the table indicate the number of problems where the compared algorithm was significantly better than, significantly worse than, or statistically equivalent to DG-MOEA/D.

4.4. Comparative Analysis of Experimental Results

To compare the performance of the algorithms,

Table 2. This table shows the average IGD and HV values each algorithm achieves on the DTLZ test function.

Problem		IGD/HV Mean
		MOEA/D	MOEA/D-DU	NSGA-II	NSGA-III	SPEA2	IM-C-MOEA/D	LERD	DG-MOEA/D
DTLZ 1	IGD	4.11 × 102=	2.29 × 10−2=	1.28 × 10−1-	2.27 × 10−2=	2.36 × 10−2=	2.27 × 101-	8.84 × 100-	3.97 × 10−2
	HV	7.87 × 101-	8.30 × 10−1=	6.48 × 10−1-	8.31 × 10−1=	8.29 × 10−1=	0.00 × 100-	1.36 × 10−1-	8.41 × 10−1
DTLZ 2	IGD	5.45 × 102=	5.45 × 10−2=	7.32 × 10−2=	5.45 × 10−2=	5.70 × 10−2=	1.20 × 10−1-	6.64 × 10−2=	5.05 × 10−2
	HV	5.60 × 101=	5.60 × 10−1=	5.28 × 10−1=	5.60 × 10−1=	5.53 × 10−1=	4.70 × 10−1-	5.35 × 10−1=	5.60 × 10−1
DTLZ 3	IGD	1.63 × 10−1-	6.03 × 10−2=	8.68 × 10−2=	6.15 × 10−2=	6.37 × 10−2=	8.61 × 101-	1.46 × 101-	6.03 × 10−2
	HV	4.67 × 10−1-	5.43 × 10−1=	4.83 × 10−1-	5.29 × 10−1=	5.36 × 10−1=	0.00 × 100-	1.52 × 10−1-	5.54 × 10−1
DTLZ 4	IGD	2.87 × 10−1-	5.45 × 10−2+	1.30 × 10−1+	1.52 × 10−1+	2.51 × 10−1=	1.19 × 10−1+	2.93 × 10−1-	2.42 × 10−1
	HV	4.46 × 10−1=	5.60 × 10−1+	4.99 × 10−1=	5.16 × 10−1+	4.70 × 10−1=	4.75 × 10−1=	4.45 × 10−1=	4.68 × 10−1
DTLZ 5	IGD	3.39 × 10−2=	2.87 × 10−2=	6.39 × 10−3=	1.33 × 10−2=	4.86 × 10−3=	3.36 × 10−2=	2.40 × 10−2=	3.89 × 10−3
	HV	1.82 × 10−1=	1.87 × 10−1=	1.99 × 10−1=	1.93 × 10−1=	1.99 × 10−1=	1.74 × 10−1=	1.89 × 10−1=	2.02 × 10−1
DTLZ 6	IGD	3.39 × 10−2=	3.31 × 10−2=	6.68 × 10−3=	1.95 × 10−2=	4.49 × 10−3=	7.05 × 100-	2.39 × 10−2=	3.39 × 10−3
	HV	1.82 × 10−1=	1.83 × 10−1=	1.99 × 10−1=	1.91 × 10−1=	2.00 × 10−1=	0.00 × 100-	1.90 × 10−1=	2.00 × 10−1
DTLZ 7	IGD	1.76 × 10−1-	3.70 × 100-	8.04 × 10−2=	7.61 × 10−2=	6.59 × 10−2=	1.58 × 10−1-	7.34 × 10−1-	6.38 × 10−2
	HV	2.55 × 10−1=	7.75 × 10−2-	2.66 × 10−1=	2.69 × 10−1=	2.75 × 10−1=	2.47 × 10−1=	2.07 × 10−1-	2.75 × 10−1
+/-/=		0/5/9	2/2/10	1/3/10	2/0/12	0/0/14	1/9/4	0/7/7

presents the average IGD and HV metric results for the proposed algorithm and seven other multi-objective optimization algorithms under the DTLZ test function, with the decision variable dimension D set to 20.

Table 3. This table shows each algorithm’s average IGD and HV values on the DTLZ test function.

Problem		IGD/HV Mean
		MOEA/D	MOEA/D-DU	NSGA-II	NSGA-III	SPEA2	IM-C-MOEA/D	LERD	DG-MOEA/D
LSMOP1	IGD	2.036 × 10−1-	2.182 × 10−1-	3.132 × 10−1-	2.416 × 10−1-	2.638 × 10−1-	7.928 × 10−1-	4.798 × 10−1-	1.390 × 10−1
	HV	6.021 × 10−1=	5.670 × 10−1=	4.456 × 10−1-	5.146 × 10−1-	5.267 × 10−1-	7.637 × 10−2-	2.078 × 10−1-	6.050 × 10−1
LSMOP2	IGD	1.655 × 10−1-	1.062 × 10−1=	1.874 × 10−1-	1.458 × 10−1=	1.620 × 10−1-	1.649 × 10−1-	1.744 × 10−1-	1.061 × 10−1
	HV	6.884 × 10−1=	6.767 × 10−1=	6.406 × 10−1-	6.908 × 10−1=	6.893 × 10−1=	6.340 × 10−1-	6.647 × 10−1=	7.080 × 10−1
LSMOP3	IGD	6.155 × 10−1-	5.995 × 10−1-	1.184 × 100-	8.339 × 10−1-	7.209 × 10−1-	3.318 × 100-	5.346 × 100-	5.439 × 10−1
	HV	2.068 × 10−1=	1.589 × 10−1-	6.014 × 10−2-	8.510 × 10−2-	1.517 × 10−1-	0.000 × 100-	2.121 × 10−2-	2.198 × 10−1
LSMOP4	IGD	3.972 × 10−1-	2.277 × 10−1+	4.673 × 10−1-	4.018 × 10−1-	3.870 × 10−1=	3.821 × 10−1=	3.609 × 10−1=	3.429 × 10−1
	HV	3.594 × 10−1=	5.969 × 10−1+	2.832 × 10−1-	4.393 × 10−1+	3.631 × 10−1=	3.038 × 10−1=	4.196 × 10−1+	3.464 × 10−1
LSMOP5	IGD	8.023 × 10−1=	2.634 × 10−1+	3.205 × 10−1+	3.117 × 10−1+	3.831 × 10−1+	7.030 × 10−1+	3.689 × 10−1+	8.241 × 10−1
	HV	9.806 × 10−2=	2.158 × 10−1+	2.298 × 10−1+	1.992 × 10−1+	2.978 × 10−1+	5.065 × 10−2-	3.549 × 10−1+	9.750 × 10−2
LSMOP6	IGD	9.645 × 10−1-	1.120 × 100-	1.319 × 100-	1.714 × 100-	1.521 × 100-	4.231 × 100-	1.869 × 100-	9.116 × 10−1
	HV	7.371 × 10−3=	0.000 × 100=	0.000 × 100=	2.334 × 10−4=	0.000 × 100=	0.000 × 100=	1.376 × 10−4=	1.663 × 10−2
LSMOP7	IGD	8.130 × 10−1-	6.937 × 10−1=	2.073 × 100-	2.436 × 100-	2.239 × 100-	2.021 × 100-	1.208 × 100-	6.526 × 10−1
	HV	8.875 × 10−2=	8.955 × 10−2=	0.000 × 100-	0.000 × 100-	0.000 × 100-	0.000 × 100-	2.898 × 10−2-	9.089 × 10−2
LSMOP8	IGD	7.966 × 10−1=	2.366 × 10−1+	3.548 × 10−1+	3.406 × 10−1+	3.530 × 10−1+	4.030 × 10−1+	3.622 × 10−1+	8.158 × 10−1
	HV	8.109 × 10−2=	2.984 × 10−1+	3.228 × 10−1+	2.907 × 10−1+	3.121 × 10−1+	1.627 × 10−1+	3.303 × 10−1+	8.230 × 10−2
LSMOP9	IGD	4.765 × 10−1-	8.840 × 101-	1.334 × 100-	1.196 × 100-	1.395 × 100-	1.581 × 100-	1.828 × 100-	4.244 × 10−1
	HV	7.852 × 10−2-	0.000 × 100-	1.030 × 10−1=	1.213 × 10−1=	1.099 × 10−1=	1.070 × 10−2-	1.125 × 10−1=	1.282 × 10−1
+/-/=		0/8/10	6/6/6	4/12/2	5/9/4	4/9/5	3/12/3	5/9/4

presents the average results of the IGD and HV metrics under the LSMOP test function, with the decision variable dimension D set to 100. All test functions in this paper are set with three objectives, and the best-ranked results are displayed in bold. Overall, the proposed method, DG-MOEA/D, demonstrates superior performance in terms of convergence and diversity.

4.4.1. Analysis of Experimental Results for the DTLZ Test Function

Based on the IGD and HV metrics presented in Table 2, the DG-MOEA/D algorithm secured 11 out of the 14 optimal results, demonstrating its exceptional optimization capability across diverse complex multi-objective problems. MOEA/D-DU achieved 2 optimal results, while NSGA-III obtained 1 optimal result; no other algorithms attained top-tier performance. DG-MOEA/D performed particularly well on the DTLZ2, DTLZ3, DTLZ5, DTLZ6, and DTLZ7 problems. The following sections analyze algorithmic performance in relation to the characteristics of each test problem.

Problem DTLZ1 presents a straightforward linear Pareto front, devoid of conflicting objective functions, serving as a benchmark for assessing algorithmic convergence and fundamental performance. In this context, NSGA-III attained the best IGD outcome. The linear uniform distribution of the Pareto frontier is consistent with NSGA-III’s reference point distribution strategy, thereby supporting the preservation of solution diversity along the front.

In the case of DTLZ4, where objective values are significantly affected by a limited number of key variables and the Pareto frontier demonstrates an uneven distribution, the MOEA/D-DU algorithm demonstrates superior IGD and HV values. This superiority is attributed to its distance-based dynamic weight adjustment mechanism: by adaptively reallocating search resources, the algorithm prevents solution loss in peripheral regions, thereby ensuring a more uniform distribution of solutions across the objective space. This attribute facilitates effective coverage of both the densely populated central region and the sparsely distributed peripheral areas in DTLZ4, leading to enhanced HV metrics relative to other algorithms.

DTLZ2 requires solutions to be uniformly distributed across a spherical frontier, whereas DTLZ3 introduces a multimodal challenge characterized by non-linear coupling among objectives. DG-MOEA/D achieves optimal IGD and HV values on DTLZ2, while markedly surpassing other algorithms in HV performance on DTLZ3. Its success is attributed to strategies that guide search directions via weight vectors and maintain diversity through external archives. These approaches enable DG-MOEA/D to achieve more comprehensive coverage of the Pareto frontier, even when addressing complex nonlinear problems.

In the DTLZ5 and DTLZ6 scenarios, where the Pareto frontier degenerates into low-dimensional manifolds and traditional algorithms tend to converge prematurely, DG-MOEA/D again demonstrates robust performance. The algorithm successfully avoids local optima traps by identifying key variables through differential grouping and focusing on effective search regions via MOEA/D’s decomposition strategy, ultimately achieving optimal IGD and HV values.

DTLZ7 presents a challenging scenario characterized by multiple discrete hyperplanes and deceptive gradients. DG-MOEA/D’s grouping strategy effectively prevents futile searches, while its external archiving mechanism preserves solutions across different regions, thereby enabling the algorithm to maintain a balance between global exploration and local exploitation. As a result, the algorithm also attains the optimal IGD and HV values for this problem.

4.4.2. Analysis of Experimental Results for the LSMOP Test Function

Compared to the DTLZ test function, which primarily assesses the robustness of multi-objective optimization algorithms, the LSMOP test function concentrates on evaluating computational efficiency and feasibility when handling high-dimensional decision variables.

According to the IGD and HV metrics in Table 3, DG-MOEA/D secured 12 out of 18 top results, demonstrating its formidable optimization capabilities across diverse large-scale multi-objective problems. MOEA/D-DU achieved 4 top results, while LERD obtained 2 top results; no other algorithms attained any leading performance. DG-MOEA/D particularly excelled on LSMOP1-3, LSMOP6, LSMOP7, and LSMOP9. The following sections provide an in-depth analysis of each algorithm’s performance across the distinct characteristics of the LSMOP test problems.

LSMOP1 integrates linear and non-linear objectives, assessing the adaptability of algorithms to mixed characteristics. DG-MO × A/D has demonstrated exceptional capability in managing mixed-objective features, effectively optimizing subproblems post-grouping. For LSMOP2, which involves strong variable interactions and redundant variables, the grouping strategy of DG-MOEA/D markedly improved variable screening efficiency. By decomposing the problem into smaller, more manageable subproblems, DG-MOEA/D more effectively addressed the complexity associated with variable interactions and redundancy compared to other algorithms. In the case of LSMOP3, where the Pareto front encompasses multiple local optimal regions, DG-MOEA/D delivers outstanding performance by reducing the search space through grouping and maintaining diversity via an external archive. This methodology prevents premature convergence and facilitates a more comprehensive solution space exploration.

In the LSMOP4 scenario, only a minority of variables significantly influence the objective function, with the remainder being noise or redundant. Within this problem, MOEA/D-DU achieved optimal results for both IGD and HV metrics. Its dynamic updating mechanism automatically adapts to shifts in variable importance across different regions of the decision space, employing the most suitable search strategy for each domain.

In the context of the LSMOP5 scenario, which involves optimizing simulated dynamic systems where variable correlations fluctuate based on the decision space position, MOEA/D-DU exhibited superior performance concerning the IGD metric, whereas LERD excelled in terms of the HV metric. MOEA/D-DU demonstrates enhanced convergence capabilities by dynamically adjusting weight vectors to align with the evolutionary trends of the Pareto frontier. Conversely, LERD differentiates between convergence-related and diversity-related decision variables through variable analysis and applies specialized optimization strategies to the latter. This methodological approach better maintains the distribution characteristics of the solution set, ultimately resulting in exceptional HV performance.

LSMOP6 introduces complex constraints, assessing algorithms’ search capabilities within the feasible domain. The grouping strategy and external archive mechanism of DG-MOEA/D facilitate efficient navigation of constrained search spaces and the identification of high-quality solutions. In LSMOP7, which incorporates continuous, discrete, and binary variables to emulate real-world mixed optimization problems, DG-MOEA/D consistently demonstrates robust performance. It directs the search process through weight vector decomposition and utilizes dynamic truncation of the external archive to maintain solutions across regions, thereby effectively balancing global exploration with local exploitation.

LSMOP8 demonstrates significant nonlinear coupling among objective functions, which requires sophisticated model approximation or collaborative optimization. The grouping technique employed by DG-MOEA/D assists in identifying critical variable clusters, facilitating a more targeted and efficient search process.

LSMOP9 introduces stochastic perturbations into objective functions to assess algorithms’ robustness under uncertain conditions. DG-MOEA/D’s ability to discern essential variable groups via differentiated clustering allows it to attain optimal results on IGD and HV metrics, even in environments affected by noise.

4.5. Ablation Experiment

The primary objective of this experimental design is to validate the effectiveness of three core innovations. The benchmark functions used for testing remain the DTLZ and LSMOP test functions from Section 4.4. DG-MOEA/D1 removes the differential grouping module, randomly assigning variables to subproblems, to assess the impact of differential grouping on solution quality and convergence speed. DG-MOEA/D2 retains DG grouping but removes external archiving, using only the current population as the solution set, to validate the role of external archiving in maintaining diversity and elite retention. DG-MOEA/D3 uses traditional differential grouping to replace spectral clustering grouping, demonstrating the superiority of improved spectral clustering under complex dependencies. The comparison results with the DG-MOEA/D algorithm are shown in

Table 4. This table shows the results of the ablation experiment.

Problem	IGD Mean
	DG-MOEA/D1	DG-MOEA/D2	DG-MOEA/D3
DTLZ1	4.57 × 10−2	3.30 × 101	2.16 × 101
DTLZ2	7.45 × 10−2	3.27 × 10−1	5.66 × 10−2
DTLZ3	6.96 × 10−2	1.91 × 102	7.92 × 101
DTLZ4	4.11 × 10−1	3.53 × 10−1	7.09 × 10−1
DTLZ5	5.39 × 10−2	1.87 × 10−1	3.04 × 10−2
DTLZ6	5.39 × 10−2	1.37 × 101	1.77 × 100
DTLZ7	1.54 × 10−1	5.34 × 10−1	1.73 × 10−1
LSMOP1	6.02 × 10−1	5.67 × 10−1	4.46 × 10−1
LSMOP2	7.08 × 10−1	7.27 × 10−1	6.41 × 10−1
LSMOP3	2.07 × 10−1	1.59 × 10−1	6.01 × 10−2
LSMOP4	3.59 × 10−1	5.97 × 10−1	3.93 × 10−1
LSMOP5	8.98 × 10−1	1.22 × 100	1.23 × 100
LSMOP6	1.01 × 100	2.50 × 100	2.26 × 100
LSMOP7	6.89 × 10−1	1.09 × 100	1.97 × 100
LSMOP8	8.21 × 10−1	2.30 × 100	1.82 × 100
LSMOP9	4.99 × 10−1	9.53 × 10−1	1.10 × 100
+/−/=	1/9/6	1/15/0	1/13/2

, where the symbols ‘+’, ‘−’, and ‘=’ indicate that the compared algorithm is significantly better than, significantly worse than, or equivalent to DG-MOEA/D, respectively.

The experimental results show that DG-MOEA/D2 performs poorly on some test problems mainly because, after removing the external archive mechanism, the algorithm cannot consistently keep historical non-dominated solutions and depends only on the current population for selection, which causes premature convergence. This is not due to over-reliance on the archive but reflects the critical importance of the archive mechanism in maintaining diversity in problems with significant objective conflicts. Additionally, while differential grouping can enhance convergence speed, it cannot fully replace the diversity-preserving function of archiving.

The result of the ablation experiments clearly demonstrate the necessity of the differential grouping algorithm in non-linear dependency problems, the contribution of external archives to long-term diversity, the advantages of the spectral clustering differential grouping algorithm over traditional methods, and the trade-offs between time and performance of the algorithm.

5. Application of DG-MOEA/D in Financial Asset Allocation

Financial asset allocation is a typical complex multi-objective optimization problem. Its main challenge is balancing multiple conflicting objectives, such as maximizing returns, controlling risk, ensuring liquidity, and complying with regulatory constraints, while also adapting to market uncertainties like economic cycles, policy changes, and sudden events. Large-scale asset pools can quickly lead to high computational complexity, necessitating the use of dimension reduction techniques or heuristic algorithms to boost efficiency. Diversified investments are also essential to reduce non-systemic risk and prevent false diversification, such as excessive industry concentration.

5.1. Data Sources and Preprocessing

To validate the effectiveness and feasibility of the DG-MOEA/D algorithm, this paper establishes a three-objective portfolio model and sets different parameters to explore the algorithm’s impact on the portfolio. Pre-grouping is very important in optimizing portfolios to address multi-objective portfolio optimization problems and overcome the difficulty of selecting stocks in the early stages of portfolio optimization model construction. Based on the characteristics of asset returns, risk, and volatility trends, this paper uses differential grouping to classify assets into two major categories: completely separable and other assets. Additionally, the similarity matrix is recalculated for grouping partially separable and non-separable assets to ensure high intra-group asset correlation and low inter-group correlation. Furthermore, the proposed DG-MOEA/D is compared with six other multi-objective evolutionary algorithms, and the Pareto optimal sets obtained by the algorithms are analyzed and compared.

This paper utilizes data from the Choice Financial Data Platform, gathering transaction data from 1 January 2023 to 31 December 2024, covering closing price information for 484 trading days. This dataset is currently considered “~large”—meaning it can be readily loaded into a single machine’s memory, processing time is reasonable, and there is no need for complex parallel or distributed systems. The selection of this dataset primarily aims to validate the algorithm’s feasibility within the financial asset allocation domain, establishing a foundation for subsequent research. Moreover, in future research, datasets with extended time spans may also be employed to further assess the robustness and scalability of the algorithm.

This study selected 23 financial assets with industry representativeness as research samples, as shown in

Table 5. This table shows the names of financial assets and their corresponding codes.

Asset Name	Asset Code	Asset Name	Asset Code	Asset Name	Asset Code
Only Education	600661.SH	China Gold	600916.SH	Contemporary Amperex Technology	300750.SZ
Baoxin Software	600845.SH	Ninghu Expressway	600377.SH	Pudong Development Bank	600000.SH
China Vanke	000002.SZ	Shanghai Airport	600019.SH	Bosera Gold ETF Link C	002611.OF
Tong Ren Tang	600085.SH	Longping High-Tech	000998.SZ	E Fund Everlasting Bond A	000265.OF
Sinopharm Group	600420.SH	Wufangzhai Industry	603237.SH	Bosera Enjoyment Holding Period A	000783.OF
Black Peony	600510.SH	Everbright Securities	601788.SH	CMF Wealth Management Bond A	000808.OF
Industrial Bank	601166.SH	PetroChina Company	601857.SH	GF Jingming Bond A	006591.OF
Edifier Technology	002351.SZ	Gansu Energy Chemical	000552.SZ

, which are widely distributed across multiple industries and fields, including education, energy, finance, pharmaceuticals, food, and new energy, ensuring the universality and reliability of the research results. Figure 2 shows the correlation coefficient heatmap between financial assets, which is a visual representation of a similarity matrix. This heat map intuitively presents the interrelationships between assets, helping investors identify risk hedging opportunities and construct cross-asset strategies.

Given the wide distribution of closing price data and the potential for outliers caused by numerical differences, this paper employs a normalization method to pre-process daily closing price data for financial assets to improve model training efficiency and prediction accuracy. Normalization involves applying specific mathematical transformations to map the original data to the [0, 1] interval. This not only effectively limits the range of data fluctuations but also significantly accelerates the convergence speed of subsequent optimization algorithms and may improve the accuracy of model solutions to some extent. It can be assured that we only perform a linear transformation on the data, preserving the relative relationships of the original data. This processing method does not introduce additional bias or alter the distribution characteristics of the data. The specific normalization formula is as follows:

$$X_i^{\mathrm{\prime }}={\displaystyle \frac{X_i-X_{min}}{X_{max}-X_{min}}}$$

(13)

where $X_i$ represents the original closing price of a stock on the $i$-th trading day, $X_{min}$ and $X_{max}$ denote the minimum and maximum closing prices of the stock during the entire observation period, respectively, while $X_i^{\mathrm{\prime }}$ refers to the closing price data after normalization.

5.2. Performance Comparison of Algorithms

5.2.1. Comparison and Analysis of Model Evaluation Metrics

Based on the three-objective portfolio model described in Section 3.5 and the DG-MOEA/D proposed in this paper, all algorithms were tested through 30 independent repetitions of experiments under the same computational environment, with the same initial population set for each algorithm to ensure fairness in the experiments. Additionally, performance evaluation metrics were used to measure and compare the performance of each algorithm.

Table 6. The table shows the average, best, and worst values of the IGD and HV indicators for the seven algorithms.

Algorithm	Average Value		Best Value		Worst Value
	IGD	HV	IGD	HV	IGD	HV
NSGA-II	8.9354 × 10−1	3.4524 × 10−1	4.4742 × 10−1	4.4524 × 10−1	8.7129 × 10−1	2.1210 × 10−1
NSGA-III	2.5827 × 10−1	1.9652 × 10−1	2.5577 × 10−1	2.2652 × 10−1	3.7981 × 10−1	1.6889 × 10−1
SPEA2	5.1803 × 10−1	3.1955 × 10−1	5.1803 × 10−1	3.4955 × 10−1	9.1152 × 10−1	2.0136 × 10−1
MOEA/D	9.6598 × 10−1	2.3837 × 10−1	6.2494 × 10−1	2.8837 × 10−1	6.9913 × 10−1	1.5555 × 10−1
IM-C-MOEA/D	3.6480 × 100	3.5529 × 10−1	2.4600 × 100	5.4945 × 10−1	5.4636 × 100	3.4846 × 10−1
LERD	3.9411 × 10−1	2.0537 × 10−1	4.9534 × 10−1	2.0537 × 10−1	4.1792 × 10−1	2.0535 × 10−1
DG-MOEA/D	2.0025 × 10−1	4.0549 × 10−1	2.9327 × 10−1	5.1627 × 10−1	8.3399 × 10−1	2.0830 × 10−1

presents the average, best, and worst values for the IGD and HV metrics, showing that DG-MOEA/D performs well across these metrics. Figure 3 shows the radar charts of IGD and HV values for several comparison algorithms. Although DG-MOEA/D is slightly inferior regarding the worst value of the IGD metric, its performance in other aspects is significantly better than that of the comparison algorithms. This indicates that the proposed algorithm has advantages in both convergence and diversity.

5.2.2. Selection and Analysis of the Optimal Investment Portfolio

After solving the multi-objective optimization stage, the Pareto frontier of the investment portfolio is obtained, as shown in Figure 4. As can be observed in the figure, the asset allocation strategy proposed in this paper achieves maximum returns while maintaining the same risk level. The optimal solution on the Pareto frontier indicates that the risk level cannot be further reduced without sacrificing returns, suggesting a positive correlation between returns and risk. Further analysis of the behavioral patterns of investors with different risk preferences reveals that investors seeking high returns tend to concentrate their investments in a small number of stocks with higher yields, thereby assuming higher risks. Risk-averse investors, on the other hand, tend to adopt a diversified investment strategy, spreading their investments across multiple stocks, particularly those with relatively lower expected returns but better risk diversification effects, thereby reducing risk.

As shown in Figure 4, there is a positive correlation between returns and risk. Clearly, this pattern aligns with the basic expectations of financial theory. Therefore, this paper assigns corresponding weights to each financial asset through an algorithm to find the optimal balance between returns and CVaR risk. Ultimately, an ideal investment portfolio solution was obtained, with the specific assets and their weights listed in descending order as shown in

Table 7. This table displays the weightings of each asset in the optimal portfolio.

Asset Code	Asset Weighting	Asset Code	Asset Weighting
600661.SH	29.375%	000808.OF	1.973%
000002.SZ	24.050%	600019.SH	1.489%
600845.SH	17.187%	002351.SZ	1.485%
600000.SH	6.409%	601857.SH	0.518%
000265.OF	5.332%	600510.SH	0.250%
601788.SH	4.290%	000998.SZ	0.234%
000783.OF	2.652%	000552.SZ	0.152%
006591.OF	2.556%

. Assets with weights less than 0.1% were excluded from actual capital allocation to ensure the concentration and efficiency of the investment portfolio.

6. Conclusions

This paper introduces a modified MOEA/D algorithm, designated DG-MOEA/D, which integrates differential grouping with an external archiving mechanism. It is evaluated against the MOEA/D, MOEA/D-DU, SPEA2, NSGA-II, NSGA-III, IM-C-MOEA/D, and LERD algorithms using the DTLZ1–DTLZ7 and LSMOP1–LSMOP9 test problems, and its application to a financial asset allocation problem is also explored. Through comparative analysis employing the IGD and HV metrics, the effectiveness of the proposed DG-MOEA/D algorithm in enhancing population diversity and distribution has been substantiated, with its benefits becoming particularly evident in large-scale multi-objective optimization challenges. In financial asset allocation across multiple industries, the DG-MOEA/D algorithm has been validated via the IGD and HV metrics, demonstrating a greater diversity of asset allocation combinations than the other seven multi-objective evolutionary algorithms. This affirms the feasibility of the proposed algorithm for practical implementation.

Although DG-MOEA/D demonstrated satisfactory convergence and distributivity in this study, its scalability and robustness in real-world financial scenarios remain constrained by two limitations. Firstly, the RDGSC stage requires constructing and storing an O($n$²) similarity matrix alongside feature decomposition. When the number of assets increases to several thousand, single-node memory and CPU overhead escalate significantly, leading to substantial growth in both runtime and space complexity. Secondly, the model employed in this research is relatively simplistic, failing to adequately account for the instability and rapid fluctuations inherent in real financial markets. The model lacks embedded mechanisms for online responses to regime shifts and tail risks: when markets transition into new regimes characterized by heightened volatility and synergy due to liquidity crises, policy abruptions, or black swan events, historical interaction graphs may become obsolete. This could lead to simultaneous sharp declines across originally grouped assets, with CVaR potentially underestimating extreme losses. Future research will explore strategies balancing algorithmic search quality with convergence efficiency to better address the core challenges of large-scale, real-time asset allocation.