A Multi-Objective Optimization Framework for Low-Carbon Index Construction and Application in Green Finance

The multi-objective optimization (MOO) framework serves as the foundation for addressing conflicting goals in low-carbon index construction, such as minimizing carbon emissions and maximizing financial returns. These objectives often conflict, as strategies to reduce emissions—like adopting renewable energy or green technologies—can involve substantial costs that may negatively impact financial performance. MOO provides a systematic approach to balance these competing objectives. Mathematically, the MOO problem is expressed as: (1) minF(x)=[ f1(x),f2(x) ]\[\min F(x)=\left[ {{f}_{1}}(x),{{f}_{2}}(x) \right]\] subject to: (2) gi(x)≤0,hj(x)=0,x∈X\[\begin{matrix} {{g}_{i}}(x)\le 0, & {{h}_{j}}(x)=0, & x\in X \\ \end{matrix}\] where f₁(x) represents carbon emissions, quantified through measures such as carbon intensity or total emissions, and f₂(x) corresponds to financial performance, evaluated through indicators like return on investment (ROI). The constraints g_i(x) and h_j(x) ensure feasibility within the solution space X.

The solutions to an MOO problem are not singular; instead, they form a Pareto-optimal front. A solution x* is Pareto-optimal if no other solution x exists such that f_i(x) ≤ f_i(x*) for all i, with at least one strict inequality. This ensures that each Pareto solution represents a trade-off where improving one objective worsens another. Decision-makers can use the Pareto front to identify solutions that align with their priorities.To achieve this, the proposed framework utilizes a hybrid algorithm combining genetic algorithms (GAs) and gradient-based methods. GAs are effective in exploring large and complex solution spaces globally, while gradient-based methods refine solutions locally for higher precision. The hybrid approach leverages the strengths of both techniques, ensuring a diverse and accurate Pareto front.The GA component begins with the initialization of a population of solutions randomly distributed within the feasible space X. Each solution is evaluated using a fitness function, which aggregates the objectives: (3) Ffitness (x)=αf1(x)+βf2(x)\[{{F}_{fitness~}}(x)=\alpha {{f}_{1}}(x)+\beta {{f}_{2}}(x)\] where α and β are dynamically adjusted weights. These weights ensure that both objectives are fairly represented and prevent bias toward a single objective. Over successive generations, solutions evolve through selection, crossover, and mutation. Selection uses methods like roulette wheel sampling to prioritize high-fitness solutions, while crossover combines solutions to generate diverse offspring. Mutation introduces random variations to maintain population diversity and prevent premature convergence.

After the GA phase, gradient-based refinement is applied to enhance local precision. The gradient of the combined objective function is computed, and the solution vector x is adjusted in the direction that minimizes the objectives: (4) ∇F(x)=[ ∂f1∂x1⋯∂f1∂xn∂f2∂x1⋯∂f2∂xn ]\[\nabla F(x)=\left[ \begin{matrix} \frac{\partial {{f}_{1}}}{\partial {{x}_{1}}}\cdots \frac{\partial {{f}_{1}}}{\partial {{x}_{n}}} \\ \frac{\partial {{f}_{2}}}{\partial {{x}_{1}}}\cdots \frac{\partial {{f}_{2}}}{\partial {{x}_{n}}} \\ \end{matrix} \right]\]

This step ensures that the solutions are both globally diverse and locally optimal, improving the quality of the Pareto front.The computational efficiency of the algorithm is crucial for handling large-scale datasets with high dimensionality. The complexity of the framework is expressed as: (5) O(N⋅M⋅G)\[\mathcal{O}(N\cdot M\cdot G)\] where N is the population size, M is the number of objectives, and G is the number of generations. To reduce computational time, parallel processing techniques are employed, dividing the solution space into smaller subspaces processed independently. This significantly reduces runtime without compromising accuracy.

Visualization of the Pareto-optimal front aids decision-making by illustrating the trade-offs between objectives. Each point on the front represents a feasible solution, with its coordinates corresponding to the values of f₁(x) and f₂(x). Decision-makers can prioritize solutions based on their specific goals, such as emphasizing environmental benefits or economic returns.The proposed MOO framework combines global exploration, local refinement, and computational efficiency to address the challenges of low-carbon index construction. By balancing competing objectives and integrating scalability and adaptability, the framework offers a robust solution for decision-making in green finance.

The multi-objective optimization (MOO) framework serves as the foundation for addressing conflicting goals in low-carbon index construction, such as minimizing carbon emissions and maximizing financial returns. These objectives often conflict, as strategies to reduce emissions—like adopting renewable energy or green technologies—can involve substantial costs that may negatively impact financial performance. MOO provides a systematic approach to balance these competing objectives.Mathematically, the MOO problem is expressed as:

Data integration and preprocessing are essential steps for ensuring that raw, heterogeneous data sources can be transformed into a structured format suitable for optimization. The data used in this study includes a variety of sources such as carbon emissions records, financial performance indicators, and sector-specific metrics. These data sources often vary in terms of scale, units, and quality, which necessitates robust preprocessing techniques to enable meaningful analysis and modeling.The first step in preprocessing is data normalization. Since the raw data comes from multiple sources with different ranges and units, normalization ensures that all variables are on a consistent scale, preventing any single variable from disproportionately influencing the optimization process. Min-max normalization is employed, which is mathematically defined as: (6) x′=x−min(x)max(x)−min(x)\[{{x}^{\prime }}=\frac{x-\min (x)}{\max (x)-\min (x)}\] where x′ is the normalized value, x is the original value, and min(x) and max(x) are the minimum and maximum values of the variable. This transformation rescales all values to lie between 0 and 1, making them comparable across different variables.

Next, feature extraction is performed to reduce the dimensionality of the dataset while retaining the most relevant information. High-dimensional data often contain redundancy, which can increase computational complexity and lead to overfitting. Principal Component Analysis (PCA) is utilized for dimensionality reduction. The mathematical transformation is expressed as: (7) Z=XW\[Z=XW\] where Z is the reduced dataset, X is the original data matrix, and W is the matrix of principal components derived from the covariance matrix of X. PCA identifies the directions (principal components) that capture the maximum variance in the data, enabling the retention of essential patterns while reducing noise.

In addition to normalization and feature extraction, missing data handling is critical. Real-world datasets often contain incomplete records due to reporting errors or missing entries. To address this, missing values are imputed using statistical methods such as mean or median substitution. Alternatively, advanced techniques like K-Nearest Neighbors (KNN) imputation or matrix factorization can be applied for more accurate estimations based on the relationships between variables.Outlier detection and removal are also important preprocessing steps. Outliers can distort statistical analyses and optimization outcomes, leading to biased results. Z-score analysis is used to detect outliers, where any data point with a Z-score greater than a predefined threshold is considered an outlier. The Z-score is calculated as: (8) Zi=xi−μσ\[{{Z}_{i}}=\frac{{{x}_{i}}-\mu }{\sigma }\] where x_i is the data point, μ is the mean of the variable, and σ is the standard deviation. Outliers are either removed or replaced with values that fall within the acceptable range, ensuring that the dataset remains consistent and reliable.

Another critical aspect of preprocessing is data partitioning. The dataset is divided into subsets for parallel processing, which significantly enhances computational efficiency. Partitioning is done by splitting the data based on attributes such as time periods or geographical regions, depending on the requirements of the analysis. This allows different subsets to be processed independently, enabling faster computations without compromising accuracy.In addition to numerical preprocessing, categorical variables are transformed into numerical representations using techniques such as one-hot encoding or label encoding. For instance, if a dataset contains categorical variables such as “industry sector” or “region,” these are encoded into binary or integer values that can be used in mathematical models. The final step in preprocessing is consistency checking, which ensures that all variables align correctly across data sources. Inconsistencies, such as mismatched units or missing time frames, are resolved by standardizing units and aligning time-series data to a common reference. This ensures that the integrated dataset is coherent and ready for analysis.

Figure 1 illustrates the complete data preprocessing pipeline, showing the flow from raw data to a structured and normalized dataset ready for optimization. The pipeline highlights key steps, including normalization, feature extraction, outlier handling, and partitioning. The comprehensive data preprocessing approach ensures that the input to the optimization framework is not only accurate but also computationally efficient. By addressing issues such as scale discrepancies, missing values, and redundancy, the preprocessing pipeline lays the groundwork for reliable and meaningful optimization outcomes.

Figure 1.

Data preprocessing workflow from raw input to structured dataset.

The low-carbon index is a composite measure designed to evaluate and quantify the balance between environmental sustainability and financial performance. This index serves as a key output of the optimization framework, integrating multiple criteria to provide a comprehensive assessment of carbon efficiency and economic viability.The construction of the low-carbon index is based on a weighted sum model, which aggregates several key metrics relevant to both carbon emissions and financial returns. Mathematically, the index is expressed as: (9) Ilow−carbon =∑i=1nwiCi\[{{I}_{low-carbon~}}=\sum\nolimits_{i=1}^{n}{{{w}_{i}}{{C}_{i}}}\] where I_low−carbon represents the low-carbon index, C_i denotes the i-th criterion (e.g., carbon intensity, renewable energy usage, financial growth), and w_i is the corresponding weight assigned to each criterion. These weights are determined using the Analytic Hierarchy Process (AHP), ensuring that each metric’s relative importance is consistently evaluated.

AHP begins with constructing a pairwise comparison matrix, where each element represents the relative importance of one criterion over another. The eigenvector of this matrix provides the weights, normalized to satisfy the condition: (10) ∑i=1nwi=1,wi≥0\[\begin{matrix} \sum\nolimits_{i=1}^{n}{{{w}_{i}}=1,} & {{w}_{i}}\ge 0 \\ \end{matrix}\]

This ensures that the index is dimensionless and interpretable, allowing comparisons across sectors or time periods.To enhance the robustness of the index, sensitivity analysis is performed to evaluate how variations in w_i affect the index outcomes. This analysis identifies critical metrics whose weights significantly impact the index, guiding policymakers or stakeholders in refining their priorities.In addition to weight determination, the selection of criteria (C_i) is crucial. The criteria are chosen to balance environmental and financial dimensions comprehensively. For instance, carbon intensity (carbon emissions per unit of GDP) captures environmental efficiency, while return on investment (ROI) measures financial profitability. Renewable energy adoption rates and energy efficiency improvements are also commonly included metrics, reflecting the transition toward sustainable practices.The constructed low-carbon index can be applied across various domains, such as evaluating the sustainability of investment portfolios, assessing sectoral carbon efficiency, or guiding policy decisions. It provides a quantitative basis for comparing different strategies or scenarios, enabling informed decision-making in green finance. By integrating both environmental and economic factors, the low-carbon index serves as a powerful tool for promoting sustainability in complex, multi-dimensional systems.

The algorithmic design and implementation of the proposed framework integrate global and local optimization strategies to achieve robust and efficient performance. This hybrid approach combines the exploration capabilities of genetic algorithms (GAs) with the precision of gradient-based methods, ensuring both diversity and accuracy in the solution set.The algorithm begins with the initialization of a population of candidate solutions randomly generated within the feasible solution space X. Each candidate solution is evaluated using a fitness function that aggregates the multiple objectives of the optimization framework.

The genetic algorithm component performs global exploration of the solution space through iterative steps. Candidate solutions are selected for reproduction using methods like roulette wheel sampling or tournament selection. Solutions with higher fitness scores are more likely to be selected, guiding the population toward optimal regions of the solution space. Parent solutions are combined through crossover operations to produce offspring, introducing new combinations of features and enhancing diversity. Random changes are introduced into offspring solutions through mutation, preventing premature convergence and maintaining genetic diversity.Once the genetic algorithm identifies a set of promising solutions, a gradient-based refinement process is applied to improve their precision. A key feature of the algorithm is its adaptability to dynamic environments. Real-world scenarios in green finance often involve real-time updates to carbon trading prices, financial metrics, or policy changes. The proposed algorithm incorporates dynamic adaptation mechanisms that recalibrate the objective functions and constraints as new data becomes available. This ensures that the solutions remain relevant and actionable, even as conditions evolve.The convergence of the algorithm is guaranteed through its hybrid design. The genetic algorithm ensures global exploration, while the gradient-based method accelerates convergence to the Pareto-optimal front.

Figure 2 illustrates the hybrid algorithm’s flow, combining the GA component for exploration and the gradient-based method for refinement. The diagram highlights the iterative steps, from initialization to the generation of Pareto-optimal solutions.

Figure 2.

Flowchart of the hybrid algorithm, combining genetic algorithms and gradient-based refinement.

The integration of these components makes the proposed algorithm highly effective in balancing exploration and exploitation, addressing the challenges of large-scale, multi-objective optimization in dynamic and complex environments. By leveraging its hybrid structure and adaptability, the algorithm provides a robust foundation for constructing low-carbon indices and supporting sustainable decision-making.