A Constraint-Tightening Feasible-Trajectory-Guided Two-Stage Evolutionary Algorithm

Abstract

Constrained multi-objective optimization problems (CMOPs) are widely encountered in practical engineering and scientific applications. To address these issues, this paper proposes a Constraint-Tightening Feasible-Trajectory-Guided Two-Stage Evolutionary Algorithm (CT-FTREA). By dividing the optimization process into a feasible-region-guided stage and a constrained Pareto front (CPF)-focused search stage, the algorithm effectively improves search efficiency and solution quality under complex constraints. In the first stage, CT-FTREA introduces an adaptive constraint boundary tightening strategy based on the number of function evaluations. By gradually reducing the $\epsilon$-constraint boundaries, the population is guided from a relaxed search space toward the feasible region. This stage also employs objective-space reference points and a weighted fitness evaluation mechanism to select and evolve individuals, while an elite archive strategy preserves obtained feasible solutions, thereby enhancing the population’s ability to advance toward the feasible Pareto front. In the second stage, CT-FTREA exchanges the roles of the population and the archive, shifting the search focus to the fine-grained approximation of the CPF. An improved elite selection strategy combined with a differential evolution operator is used to generate offspring, adaptively balancing the exploration and exploitation capabilities of the population. The computational complexity of CT-FTREA is $O(F{E}_{max}\times N)$, where $F{E}_{max}$ is the maximum number of function evaluations and N is the population size. Extensive experiments on 28 benchmark instances and four real-world engineering problems show that CT-FTREA outperforms seven state-of-the-art algorithms. Specifically, it achieves the best IGD result with a 54% improvement and the best HV result with a 50% improvement over competing methods on the test problems. The algorithm also demonstrates statistically significant advantages in terms of convergence, solution quality, and robustness (Wilcoxon rank-sum test at a 0.05 significance level). CT-FTREA algorithm offers an efficient and robust solution to CMOPs, with competitive performance on both benchmark and real-world problems.

1. Introduction

Constrained multi-objective optimization problems (CMOPs) are widely encountered in practical engineering and scientific applications, including energy management [1], transportation systems [2], industrial process optimization [3], and control engineering [4,5]. Unlike unconstrained multi-objective optimization, CMOPs require simultaneous optimization of multiple conflicting objectives while satisfying a set of complex constraints, which significantly increases the difficulty of the search process.

One of the major challenges in CMOPs lies in the irregular and disconnected feasible regions, where feasible solutions may occupy only a small portion of the search space [6,7]. Moreover, the true CPF is often surrounded by a large number of infeasible solutions, making it difficult for evolutionary algorithms to effectively approach and approximate the CPF [8,9,10]. As a result, conventional multi-objective evolutionary algorithms (CMOEAs) frequently suffer from slow convergence and poor solution diversity when directly applied to CMOPs.

To formally describe CMOPs, a general constrained multi-objective optimization problem can be defined as

$$\begin{array}{cc}\hfill \underset{X\in \Omega }{min} & F\left(X\right)=\left(f_1\left(X\right),f_2\left(X\right),\dots ,f_m\left(X\right)\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}. & g_i\left(X\right)\le 0, i=1,\dots ,p,\hfill \\ \hfill & h_j\left(X\right)=0, j=1,\dots ,q,\hfill \end{array}$$

(1)

where $F\left(X\right)$ denotes the objective vector, $\Omega$ represents the decision space, and $g_i\left(X\right)$ and $h_j\left(X\right)$ correspond to inequality and equality constraints, respectively.

In such problems, the difficulty of search is largely attributed to the complex constraint landscape and the scarcity of feasible solutions. To quantitatively characterize the degree of infeasibility, the violation of each constraint is defined as

$$C_j\left(X\right)=\left\{\begin{array}{cc}max\{0, g_j\left(X\right)\},\hfill & j=1,\dots ,p,\hfill \\ max\{0, |h_{j-p}\left(X\right)|-\delta \},\hfill & j=p+1,\dots ,p+q,\hfill \end{array}\right.$$

(2)

where $\delta$ denotes a small tolerance for equality constraints. Accordingly, the overall constraint violation of a solution X is calculated as

$$CV\left(X\right)=\sum _{j=1}^{p+q}{C}_j\left(X\right).$$

(3)

The above formulation enables a continuous evaluation of feasibility, which provides a basis for advanced constraint-handling mechanisms [11]. In particular, it facilitates the design of strategies that progressively guide the population from infeasible regions toward feasible regions [12], rather than enforcing feasibility in a single step.

To address these challenges, various constraint-handling techniques have been proposed, including penalty-based methods [13], feasibility rule-based approaches [14], and $\epsilon$-constraint methods [15,16]. Among them, the $\epsilon$-constraint method has attracted considerable attention due to its ability to balance constraint satisfaction and objective optimization by allowing controlled constraint violations during early search stages [17,18]. Recent studies further demonstrate that adaptive constraint relaxation strategies can significantly enhance feasibility discovery and convergence performance in complex CMOPs [19,20].

In parallel, multi-stage [21,22] and adaptive evolutionary frameworks [23,24] have emerged as effective solutions for handling complex CMOPs. By dividing the optimization process into different stages with distinct search objectives, such algorithms can guide populations from infeasible regions toward feasible regions and subsequently focus on CPF refinement [25,26]. Archive-assisted strategies [27,28] and differential evolution (DE) operators have also been widely integrated to enhance convergence accuracy and robustness under constrained environments.

Motivated by these developments, this paper proposes a Constraint-Tightening Feasible-Trajectory-Guided Two-Stage Evolutionary Algorithm (CT-FTREA), which explicitly integrates adaptive constraint tightening, feasible-trajectory guidance, and archive–population role interaction to efficiently address complex CMOPs.

The main contributions of this paper are summarized as follows:

(1)

This paper proposes a novel Constraint-Tightening Feasible-Trajectory-Guided Two-Stage Evolutionary Algorithm (CT-FTREA) for constrained multi-objective optimization problems. The proposed framework decomposes the optimization process into a feasible-region-guided exploration stage and a CPF-oriented refinement stage. In the first stage, an adaptive $\epsilon$-constraint boundary tightening strategy is designed to progressively guide the population from relaxed infeasible regions toward the feasible region by dynamically shrinking the constraint boundary according to the number of function evaluations. This mechanism effectively exploits feasible search trajectories and enhances convergence efficiency and robustness under complex constraint environments.

(2)

A role-exchanging archive–population collaboration mechanism is introduced in the second stage, where the archive and population exchange their roles to actively guide offspring generation. Combined with an improved elite selection strategy and a differential evolution operator, this mechanism enables fine-grained approximation of the CPF by adaptively balancing exploration and exploitation.

(3)

Extensive comparative experiments conducted on 28 benchmark test instances and four real-world engineering problems demonstrate that the proposed CT-FTREA achieves competitive and robust performance compared with seven state-of-the-art algorithms.

The remainder of this paper is organized as follows. Section 2 reviews related work and presents the research motivation. Section 3 introduces the CT-FTREA algorithm, including its framework and implementation details. Section 4 evaluates the algorithm on benchmark test suites and real-world CMOPs. Finally, Section 5 concludes the paper and discusses future research directions.

2. Related Work and Motivation

This section reviews the most relevant studies in constrained multi-objective optimization and discusses the motivation for developing the proposed approach. Specifically, we first examine existing constraint-handling techniques, multi-stage and adaptive evolutionary frameworks, and archive-assisted strategies, and then identify the limitations that inspire the design of our CT-FTREA algorithm.

2.1. Related Work

To effectively address constrained multi-objective optimization problems, numerous studies have focused on designing specialized algorithms that balance constraint satisfaction with objective optimization. In the following, we review the main categories of methods that have been widely adopted in the literature.

2.1.1. Constraint-Handling Techniques for CMOPs

Constraint-handling techniques play a crucial role in effectively solving constrained multi-objective optimization problems (CMOPs), as they directly determine whether the evolutionary search can efficiently navigate between infeasible and feasible regions. In recent years, constraint-handling strategies have evolved along several directions. Penalty-based approaches, such as Zhao et al. [29] and Chen et al. [30], incorporate constraint violations into fitness evaluation. While these methods can guide the population toward feasible regions, their performance is often highly sensitive to penalty coefficients, especially in highly constrained or irregular feasible landscapes. To avoid explicit penalty parameters, feasibility rule-based methods [6,31] prioritize feasible solutions and compare infeasible ones according to their violation degrees. Although simple and parameter-free, these methods may overly restrict exploration when feasible solutions are scarce. To further enhance robustness, $\epsilon$-constraint techniques [16,18] and adaptive variants such as ACREA [30] allow slightly infeasible solutions to participate in the search by relaxing constraint boundaries. Beyond classical strategies, dual-population and archive-based methods [31,32] and multitasking knowledge transfer frameworks [33] exploit auxiliary information or collaborative mechanisms to improve feasibility attainment and convergence in complex CMOPs. In many-objective scenarios, algorithms such as MOEA-AD [34] and SDEA [29] further enhance exploration and diversity by adaptive hierarchical sorting or self-organizing decomposition with cooperative diversity measures. Despite these advances, many existing approaches still rely on predefined or stage-dependent constraint-tightening strategies, which can lead to abrupt transitions in search dynamics, insufficient balance between feasibility and diversity, or premature convergence near the constrained Pareto front (CPF).

In contrast, the proposed CT-FTREA integrates a feedback-driven tightening mechanism within a progressive two-stage framework. Instead of static or manually scheduled constraint relaxation, the constraint pressure is dynamically regulated according to the evolutionary trajectory. This trajectory-aware adjustment enables smoother exploration–exploitation transitions, mitigates premature convergence near the CPF, and preserves population diversity across complex constrained landscapes, thus addressing key limitations observed in recent CMOP algorithms.

2.1.2. Multi-Stage and Adaptive Evolutionary Frameworks

To address the increasing complexity of constrained multi-objective optimization problems (CMOPs), multi-stage evolutionary algorithms decompose the optimization process into sequential phases, which separately guide feasibility exploration and Pareto front approximation [7,9,20]. However, existing methods generally lack mechanisms to exploit feasible search trajectories across stages, limiting the effective utilization of accumulated search knowledge, particularly in problems with fragmented or disconnected feasible regions. To overcome these limitations, several two-stage and multi-stage evolutionary algorithms have been proposed to enhance search efficiency and feasibility. For example, Zhang et al. [31] proposed a two-stage archive-based algorithm (CMOEA-TA), where the first stage relaxes constraints according to the proportion of feasible solutions and constraint violations to encourage broader exploration, and the second stage shares information between the archive and the population while embedding constraint dominance principles and angle-based selection to improve feasibility and diversity. Hao et al. [28] introduced a competition-based two-stage algorithm (CP-TSEA), in which the first stage applies an $\epsilon$-constraint boundary relaxation mechanism to improve diversity and global search capability, and the second stage uses an equal-probability competitive strategy and three-criteria ranking to select high-quality parents and maintain a balance between diversity and convergence. Additionally, Fan et al. [35] proposed a coevolutionary two-stage strategy, Sun et al. [21] developed a multi-stage algorithm with staged constraint handling and objective-guided search, and Qiao et al. [36] presented a high-dimensional constrained multi-objective optimization framework, providing scalable benchmarks and effective algorithms for complex high-dimensional problems.

Overall, two-stage approaches offer clear advantages over single-stage strategies by explicitly separating exploration and convergence phases, enabling stage-specific strategies that effectively balance exploration and convergence. Compared with more fine-grained multi-stage methods, two-stage frameworks retain the benefits of stage separation while avoiding the increased scheduling complexity and parameter sensitivity of multiple stages. Motivated by these observations, the proposed CT-FTREA adopts a progressive two-stage framework with a feedback-driven, trajectory-aware tightening mechanism, fully utilizing search trajectory information to balance exploration and feasibility, prevent premature convergence near the constrained Pareto front, and maintain population diversity.

2.1.3. Archive-Assisted and DE-Based Approaches

Elite archive strategies are widely used to preserve high-quality feasible solutions and prevent population diversity loss over generations [8,37]. When combined with reference-point-based fitness assignment or decomposition methods, archives provide a stable pool of promising solutions that enhance CPF approximation accuracy [31,38]. Differential evolution (DE) operators are often integrated into constrained evolutionary algorithms due to their strong global search capability and adaptability to high-dimensional problems [18,39]. However, balancing exploration and exploitation under dynamic constraint conditions remains challenging. Without effective interaction between archives and the evolving population, DE-based strategies may over-explore infeasible regions or converge prematurely to suboptimal feasible regions. To improve search efficiency and feasibility guidance, several two-stage and multi-stage algorithms incorporate archive-assisted mechanisms. For example, Zhou et al. [40] proposed a competitive dual-archive dual-stage strategy to enhance convergence and feasibility; Lai et al. [41] introduced a feedback-tracking constraint relaxation algorithm to guide the population along feasible trajectories; and Zhang et al. [31] developed a two-stage archive evolutionary algorithm coordinating archive usage across stages to maintain diversity and improve CPF approximation. Collectively, these studies illustrate the benefits of combining elite archives, DE operators, and multi-stage designs, while highlighting the need for integrated, trajectory-aware approaches to balance exploration and exploitation, maintain diversity, and enhance CPF approximation in complex CMOPs.

2.2. Motivation

Despite the significant progress achieved in constrained multi-objective evolutionary optimization, the performance of existing algorithms in complex scenarios is still hindered by several inherent limitations. First, most constraint-handling methods primarily emphasize feasibility enforcement, while lacking explicit mechanisms to guide populations along feasible search trajectories. Consequently, historical search information is not fully exploited, leading to slow convergence and insufficient exploration of feasible regions [16,19]. Second, many multi-stage evolutionary algorithms rely on heuristic or static stage-transition strategies that cannot adapt to the dynamically changing feasibility status of the population. This often results in premature convergence in early stages or inefficient exploitation in later stages, especially for problems with irregular and disconnected feasible regions [7,9]. Third, although elite archives are widely employed to preserve high-quality solutions, they are frequently treated as passive repositories rather than active components for guiding offspring generation, which limits the balance between exploration and exploitation and degrades the approximation accuracy of the CPF [8].

The above issues can be further illustrated through the visualization of the evolutionary search process, as shown in Figure 1. In this figure, the gray region denotes the feasible region, the white region denotes the infeasible region, and cyan circles indicate feasible solutions in the population. During the initial stage (Figure 1a,d), the population is mainly confined to local feasible regions due to limited exploration capability, making it difficult to cross infeasible barriers. In the middle stage (Figure 1b,e), the adaptive constraint-tightening strategy progressively guides individuals across infeasible regions, enabling the population to explore potential feasible trajectories and expand toward different constrained Pareto fronts (CPFs). At the termination stage (Figure 1c,f), most solutions reside within the feasible region, with a substantial proportion converging toward the leading edge of the CPF, illustrating the algorithm’s effectiveness in achieving both convergence and diversity. Notably, the top row of subfigures represents a simplified scenario, while the bottom row illustrates the algorithm’s behavior in a more complex constrained environment, highlighting the robustness of the proposed method across varying problem difficulties.

Figure 1. Evolutionary search process of CMOEA in CMOPs. Gray: feasible; white: infeasible; cyan: feasible solutions. Top: simple scenario; bottom: complex scenario. The algorithm’s exploration and adaptive constraint-tightening guide convergence toward the CPF.

However, existing $\epsilon$-constraint and penalty-based methods have been widely adopted to allow slightly infeasible solutions to participate in evolutionary searches. However, most of these approaches fail to explicitly model and exploit trajectory-guided evolution processes. Consequently, populations often struggle to overcome infeasible barriers during early search stages and lack effective refinement of the constrained Pareto front (CPF) in later stages [16,18]. In practical engineering problems, the complex interplay between constraints and multiple conflicting objectives produces a highly rugged and deceptive search landscape, where populations may become trapped in locally feasible regions or overlook regions offering superior trade-offs. Traditional strategies, such as fixed penalty coefficients or static $\epsilon$-constraint methods, are generally incapable of adapting to such dynamic environments, leading to either excessive constraint violations or overly conservative searches [28,42,43].

Moreover, the performance of these methods is highly dependent on the design of the $\epsilon$-relaxation schedule; inappropriate schedules may overly restrict exploration or allow excessive infeasible solutions, resulting in slow convergence [44]. Many approaches also rely on static or manually tuned parameters, which reduces robustness across different problem instances. Finally, existing $\epsilon$-constraint approaches typically lack mechanisms to dynamically exploit feasible trajectories, potentially causing the population to prematurely converge in suboptimal feasible regions and limiting solution diversity [45]. These shortcomings highlight the need for more adaptive, trajectory-aware constraint-handling strategies.

Motivated by the above observations and recent studies on archive-assisted multi-stage strategies [31,40,41], this paper proposes a unified evolutionary framework termed CT-FTREA. In the first stage, an adaptive constraint-tightening mechanism guides the population from relaxed infeasible regions toward feasible regions. In the second stage, a role-exchanging archive–population collaboration strategy is employed to further refine CPF approximation while maintaining a proper balance between exploration and exploitation. By explicitly integrating adaptive constraint regulation, feasible-trajectory guidance, and active archive–population interaction, CT-FTREA enhances convergence efficiency, solution diversity, and robustness in complex constrained multi-objective optimization scenarios.

3. Proposed CT-FTREA

Building on the motivation discussed above, this paper propose the CT-FTREA. The following subsection presents the overall framework of the algorithm and its key components.

3.1. The Framework of CT-FTREA

Algorithm 1 presents the framework of the proposed CT-FTREA for CMOPs. The algorithm begins by randomly initializing the main population ($P_M$) and calculating the CV for each individual. An archive population ($P_A$) is concurrently initialized to store high-quality feasible solutions, serving as a reference to guide the evolutionary search. CT-FTREA operates in two consecutive stages. In the first stage, the algorithm focuses on steering the population toward feasible regions while maintaining diversity. A dynamic $\epsilon$ boundary is computed using the ReduceBoundary function, allowing slightly infeasible solutions to participate in the evolutionary process. This ensures a smooth transition between exploration and exploitation and mitigates premature convergence. The mating pool is constructed by considering both $P_A$ and $P_M$, promoting convergence toward promising regions while preserving diversity. Offspring are generated through standard genetic operators, and the next generation is selected using an environmental selection mechanism that accounts for both objective values and $\epsilon$-adjusted constraint violations.

Algorithm 1 Framework of CT-FTREA

1: Input: N (Population size), $P_M$ (Main population), $P_A$ (Archive population), $F{E}_{max}$ (Maximum function evaluations),  2: Output: $P_F$ (Final population)  3: $P_M\leftarrow$ RandomInitialization(N)  4: $P_A\leftarrow \{x\in P\mid CV\left(x\right)=0\}$  5: while termination criterion not met (e.g., $FE<F{E}_{max}$) do  6:    if $FE<0.5·F{E}_{max}$ then  7:       $z\leftarrow min(P.objs)$  8:       $a\leftarrow 0.5\left(1-cos\left(\pi ·(1-FE/F{E}_{max})\right)\right)$  9:       $\epsilon \leftarrow$ ReduceBoundary($P_M$, $FE$, $F{E}_{max}$) 10:       $M_1\leftarrow$ MatingSelectionElite($P_M$, $P_A$, N, a) 11:       $Q_1\leftarrow$ OperatorGA(N, $M_1$) 12:       $P_A\leftarrow$ UpdateArchive($P_M$, $P_A$, $Q_1$, N) 13:       $P_M\leftarrow$ EnvironmentalSelection($P_M$, $Q_1$, N, [$P_M.cons-\epsilon$, $Q_1.cons-\epsilon$]) 14:    else 15:       if $stage\_conver=0$ then 16:          $temp\leftarrow P_M;P_M\leftarrow P_A;P_A\leftarrow temp$ 17:          $stage\_conver=1$ 18:       end if 19:       $M_2\leftarrow$ MatingSelectionElite($P_M$, $P_A$, N, ratio=0.5) 20:       $Q_2\leftarrow$ DEgenerator2($Problem$, $M_2$) 21:       $P_A\leftarrow$ UpdateArchive($P_A$, [$P_M$, $Q_2$], N) 22:       $P_M\leftarrow$ EnvironmentalSelectionElite($P_M$, $Q_2$, N) 23:    end if 24: end while 25: return  $P_F$

Once the main population ($P_M$) has largely entered feasible regions, the algorithm proceeds to Stage 2. At this stage, a one-time swap between the population and the archive ($P_A$) is performed because the archive already contains well-converged feasible solutions that approximate the leading edge of the constrained Pareto front (CPF). By letting these high-quality solutions actively guide offspring generation, Stage 2 accelerates convergence toward the CPF while maintaining diversity, providing more “active guidance” than in Stage 1. Differential evolution (DE) operators are applied to enhance global search capabilities, and $P_A$ is continuously updated to retain elite solutions. Environmental selection during this stage ensures a balance between convergence and diversity, allowing the population to evenly approximate the CPF. The archive $P_A$ is updated at every generation. After offspring are generated, the combined population and archive are evaluated based on objective values and $\epsilon$-adjusted constraint violations. The top N individuals are then retained in $P_A$, ensuring that it always contains high-quality, feasible, and diverse solutions. This allows the archive to actively guide offspring generation in Stage 2, accelerating convergence toward the constrained Pareto front while maintaining population diversity.

CT-FTREA is fundamentally different from hybrid methods such as PSO-DE and GA-Swarm, which typically rely on penalty functions or generic constraint-handling mechanisms. These methods often struggle with maintaining a good balance between exploration and exploitation, especially in high-dimensional constrained optimization problems. In contrast, CT-FTREA’s adaptive constraint boundary tightening strategy and two-stage optimization process enable it to more effectively navigate the feasible region and converge toward the CPF. Additionally, CT-FTREA integrates a dynamic archive strategy to preserve high-quality feasible solutions, which ensures both diversity and convergence throughout the search process.

By explicitly exploiting feasible trajectories, dynamically tightening constraints, and integrating $P_M$–$P_A$ collaboration, CT-FTREA achieves efficient convergence, maintains solution diversity, and provides high-quality approximations of the CPF, demonstrating robust optimization performance in complex CMOPs. For a more intuitive understanding of the algorithm flow, Figure 2 illustrates the framework of CT-FTREA.

Figure 2. Framework flowchart of CT-FTREA.

3.2. Adaptive Constraint Boundary Tightening Strategy

The proposed adaptive constraint boundary tightening strategy is designed to progressively guide the population from relaxed infeasible regions toward the feasible region during the evolutionary process. In constrained multi-objective optimization problems, individuals violating constraints can still contain valuable information for guiding the search. To exploit this information without prematurely discarding infeasible solutions, a dynamic $\epsilon$-constraint boundary is introduced, allowing slightly infeasible individuals to participate in the selection and reproduction processes.

At a given generation, the $\epsilon$-adjusted constraint violation of an individual x is defined as

$$C{V}_{\epsilon }\left(x\right)=max(CV\left(x\right)-\epsilon ,0),$$

(4)

where $CV\left(x\right)$ denotes the original constraint violation of x, and $\epsilon$ is a non-negative boundary controlling the allowable violation. Initially, $\epsilon$ is set relatively large to encourage exploration in wider infeasible regions. As the evolutionary process progresses, $\epsilon$ is gradually reduced according to the proportion of function evaluations completed:

$$\epsilon \left(FE\right)={\epsilon }_0·{\left(1-{\displaystyle \frac{FE}{F{E}_{max}}}\right)}^{\alpha },$$

(5)

The initial boundary ${\epsilon }_0$ is set based on the proportion of feasible solutions in the current population, $FE$ is the current number of function evaluations, $F{E}_{max}$ is the maximum allowed evaluations, and $\alpha$ is a control parameter determining the decay rate of the boundary, which is set to 0.04 in our experiments. The parameter $\alpha$ is introduced to regulate the balance between exploration of infeasible regions and convergence towards feasible solutions during the evolutionary process. The value of $\alpha$ is determined empirically based on preliminary experiments to ensure an appropriate trade-off between these competing goals. Specifically, $\alpha$ controls the constraint-tightening rate: a smaller value of $\alpha$ allows more infeasible solutions to participate in the evolution, thereby enhancing population diversity but potentially slowing convergence to the constrained Pareto front (CPF). On the other hand, a larger $\alpha$ emphasizes feasibility by tightening the constraints more quickly, leading to faster convergence but at the risk of reducing solution diversity. This dynamic adjustment ensures a smooth transition from exploration to exploitation, allowing the population to progressively focus on feasible regions without losing diversity.

By incorporating $\epsilon$-adjusted constraint violations into the environmental selection mechanism, the strategy maintains a balance between convergence and diversity. Early generations can explore infeasible regions to discover promising areas, while later generations gradually converge toward the feasible Pareto front (CPF), enhancing both the efficiency and robustness of the search process. This mechanism effectively exploits feasible trajectories and mitigates premature convergence, which is especially crucial in complex constrained multi-objective optimization problems.

3.3. Elite Environmental Selection Strategy

In CT-FTREA, the elite environmental selection strategy (Algorithm 2) is employed to simultaneously guide the population toward the feasible Pareto front (CPF) and preserve diversity. The selection process begins by identifying feasible individuals from the main population and performing non-dominated sorting based on objective values and convergence. When the number of feasible solutions exceeds the target population size, lower-ranked individuals are retained first, and those from the last front are carefully pruned using a distance-based truncation mechanism to avoid overcrowding and ensure uniform coverage along the CPF. When feasible solutions are insufficient to fill the population, slightly infeasible individuals with smaller constraint violations are incorporated, allowing the algorithm to explore potentially promising regions and maintain exploration capability in early generations.

Algorithm 2 Elite environmental selection strategy

1: Input: N (Population size), $P_M$ (Main population)  2: Output: $N_f$ (Feasible set), $N_I$ (Infeasible set)  3: $N_{Pf}\leftarrow$ Compute non-dominated sorting for the $P_M$ based on objectives and convergence.  4: $P_f=\{x\in P\mid CV\left(x\right)=0\}$  5: $N_f\leftarrow$ Update feasible($P_f$, N)  6: if $N_f$ < N then  7:    Sort $N_f$ by ascending $CV\left(x\right)$  8:    Select top $N-$length($N_f$) individuals and add to $N_f$.  9: end if 10: $N_{Pf}\leftarrow$ Compute non-dominated sorting for the $P_M$ based on CV and diversity. 11: $N_I=\{x\in P\mid CV\left(x\right)\ne 0\}$ 12: $N_{IF}\to$ retain only infeasible individuals in the $N_{Pf}$. 13: $N_I\leftarrow$ Update infeasible($N_I$, $N_{IF}$, N). 14: return $N_f$, $N_I$

Infeasible solutions are filtered through non-dominated sorting based on constraint violations and diversity, with only those from the first front retained for further consideration. These selected infeasible individuals are processed to ensure their number does not exceed the population size, thereby balancing feasibility and exploration. By integrating front-based ranking, feasibility prioritization, and distance-aware truncation for both feasible and infeasible individuals, the elite environmental selection mechanism ensures that the population progressively converges toward high-quality feasible regions while maintaining solution diversity. In combination with the adaptive constraint boundary tightening and archive–population collaboration strategies, this mechanism plays a central role in improving convergence efficiency, achieving uniform CPF coverage, and enhancing solution quality and robustness in complex CMOPs.

3.4. Time Complexity Analysis

The time complexity of the proposed CT-FTREA algorithm is primarily influenced by the number of iterations in the main loop, which is bounded by the maximum number of function evaluations $F{E}_{max}$, and the population size N. In the initialization phase, the main population $P_M$ and the archive population $P_A$ are initialized. The time complexity for initializing the population and the archive is $O\left(N\right)$, as it requires iterating over all individuals to either initialize or evaluate the constraint violation condition. In each iteration of the main loop, the algorithm performs several operations such as fitness evaluation, constraint relaxation, mating selection, crossover, mutation, archive updating, and environmental selection. All of these operations are typically executed in $O\left(N\right)$ time, where N is the population size. Therefore, for each iteration, the time complexity remains $O\left(N\right)$. The main loop executes at most $F{E}_{max}$ times, so the overall time complexity of the algorithm is

$$O(F{E}_{max}\times N)$$

This indicates that the computational cost of the CT-FTREA algorithm scales linearly with both the maximum function evaluations and the population size.

In summary, the overall time complexity of the CT-FTREA algorithm is $O(F{E}_{max}\times N)$. This linear relationship makes the algorithm computationally efficient, as it ensures a manageable computational cost while solving complex CMOPs.

4. Experimental Study

This section presents the experimental setup, competing algorithms, performance metrics, and the results analysis of the proposed CT-FTREA algorithm. This study is conducted on multiple benchmark test suites (MW, LIRCMOP) and four real-world constrained multi-objective optimization problems (CMOPs) [36].

4.1. Experimental Setup

The parameters of each CMOP are set as follows: MW4, MW8, MW14, and LIR-CMOP13–14 have $m=3$, while all other CMOPs have $m=2$; the MW problems have $n=15$, and the LIRCMOP problems have $n=30$. For the genetic algorithm (GA) operators, the crossover probability is set to 0.9 and the mutation probability to 0.1. In the second stage, the differential evolution (DE) operators use a scaling factor $F=0.5$ and a crossover probability $CR=0.9$, following the standard DE/rand/1/bin strategy. The parameters of each algorithm follow the settings reported in their original papers.

All algorithms are tested with a population size of $N=100$ and a maximum number of function evaluations (Max FEs) of 100,000. Each algorithm is independently executed 20 times on each test instance. All algorithms were implemented using PlatEMO 4.0 [46].

4.2. Competing Algorithms

Seven state-of-the-art constrained multi-objective evolutionary algorithms (CMOEAs) are selected for comparison: BiCO [47], TSTI [48], POCEA [49], IMTCMO [36], TPCMaO [50], MTCMO [51], and CMOEMT [33]. All competing algorithms strictly follow the parameter settings and operational strategies recommended in their original publications to ensure fairness and reproducibility. The following CMOEAs are used for comparison:

(1)

BiCO: Handling Constrained Multi-objective Optimization Problems Via Bidirectional Coevolution.

(2)

TSTI: A Two-stage Evolutionary Algorithm Based on Three Indicators for Constrained Multi-objective Optimization.

(3)

POCEA: Paired Offspring Generation for Constrained Large-scale Multi-objective Optimization.

(4)

IMTCMO: Evolutionary Constrained Multi-objective Optimization: Scalable High-dimensional Constraint Benchmarks and Algorithm.

(5)

TPCMaO: Solving Optimal Power Flow Problems Via a Constrained Many-objective Co-evolutionary Algorithm.

(6)

MTCMO: Dynamic Auxiliary Task-based Evolutionary Multitasking for Constrained Multi-objective Optimization.

(7)

CMOEMT: Constrained Multi-objective Optimization Via Multitasking and Knowledge Transfer.

4.3. Performance Metrics

The inverted generational distance (IGD⁺) [52] is used to measure the average distance between the obtained solution set and the true constrained Pareto front (CPF). Since the traditional IGD⁺ is not Pareto-compliant, the improved IGD⁺ is adopted. A smaller IGD⁺ value indicates better convergence and distribution performance.

The hypervolume (HV) [53] indicator is employed as a complementary metric, which measures the volume of the objective space dominated by the solution set with respect to a predefined reference point. A larger HV value implies superior performance.

For IGD⁺ calculation, approximately 10,000 uniformly distributed points are sampled from the true CPF as the reference set. Before computing HV, the objective values are normalized, and $(1.1,1.1,\dots ,1.1)$ is selected as the reference point. The experimental results are reported in terms of mean and standard deviation, and the Wilcoxon rank-sum test with a significance level of 0.05 is conducted for statistical analysis. The symbols “+”, “−”, and “≈” indicate that the compared algorithm performs significantly better than, significantly worse than, or statistically similar to CT-FTREA, respectively. The gray shading highlights the best performance on each CMOP test problem, while “NaN” denotes that the algorithm fails to find feasible solutions.

4.4. Experimental Results and Analysis

The performance of CT-FTREA is evaluated on the benchmark and real-world CMOPs. By comparing CT-FTREA with the seven state-of-the-art CMOEAs, its performance in terms of convergence, diversity, and feasibility is analyzed. The strengths and limitations of CT-FTREA across different constraint types and problem complexities are summarized.

4.4.1. Comparative Analysis of MW Problems

By analyzing the results reported in Table 1 and Table 2, it can be observed that CT-FTREA exhibits consistently superior overall performance in terms of both IGD⁺ and HV metrics, which highlights the effectiveness and novelty of its algorithmic design. Across most MW benchmark problems, CT-FTREA outperforms the compared algorithms in convergence accuracy and solution quality, particularly on MW1, MW2, MW5, MW6, and MW7. In these cases, the obtained solutions are able to approach the true constrained Pareto front (CPF) more closely while maintaining a well-distributed solution set. For instance, on the bi-objective MW1 and tri-objective MW14 problems, CT-FTREA achieves the smallest IGD⁺ values among all competitors, indicating its strong capability in accurately approximating the CPF. Overall statistical results further show that CT-FTREA outperforms BiCo, TSTI, POCEA, IMTCMO, TPCMaO, MTCMO, and CMOEMT on the majority of test instances, demonstrating its robustness under diverse constraint conditions. From the perspective of the HV indicator (Table 2), CT-FTREA also maintains a clear advantage, especially on problems such as MW1 and MW2, where it yields significantly larger hypervolume values than the competing algorithms. This suggests that CT-FTREA is capable of achieving a desirable balance between convergence and diversity, effectively preserving global coverage of the objective space even under high-dimensional and complex constrained scenarios.

Table 1. Statistical analysis (mean ± standard deviation) of IGD⁺ values for eight algorithms evaluated on MW benchmark problems.

Problem	M	D	BiCo	TSTI	POCEA	IMTCMO	TPCMaO	MTCMO	CMOEMT	CT-FTREA
MW1	2	15	1.3523 × 10−1 (1.08 × 10−1) −	7.9445 × 10−3 (1.84 × 10−2) ≈	1.1889 × 10−2 (3.98 × 10−3) −	1.4623 × 10−3 (1.77 × 10−4) −	3.1440 × 10−3 (6.65 × 10−3) −	8.6376 × 10−2 (7.61 × 10−2) −	1.4508 × 10−3 (3.22 × 10−4) −	1.2127 × 10−3 (3.16 × 10−5)
MW2	2	15	1.6101 × 10−2 (7.75 × 10−3) −	1.5813 × 10−2 (6.94 × 10−3) −	1.6754 × 10−1 (1.03 × 10−1) −	7.5661 × 10−2 (3.11 × 10−2) −	1.9732 × 10−2 (7.18 × 10−3) −	2.0014 × 10−2 (1.08 × 10−2) −	2.8534 × 10−2 (2.03 × 10−2) −	4.0504 × 10−3 (3.00 × 10−3)
MW3	2	15	1.8488 × 10−2 (7.66 × 10−3) −	1.1099 × 10−2 (2.09 × 10−2) ≈	3.1400 × 10−2 (1.16 × 10−2) −	1.6089 × 10−2 (6.32 × 10−3) −	2.6235 × 10−3 (3.49 × 10−4) +	7.4876 × 10−3 (1.83 × 10−3) −	3.1868 × 10−3 (4.80 × 10−4) +	3.3120 × 10−3 (1.42 × 10−4)
MW4	3	15	6.0657 × 10−2 (1.77 × 10−2) −	2.9555 × 10−2 (8.07 × 10−4) +	4.7361 × 10−2 (7.77 × 10−3) −	1.7512 × 10−1 (6.07 × 10−2) −	3.9490 × 10−2 (3.68 × 10−2) −	6.2938 × 10−2 (2.07 × 10−2) −	4.8973 × 10−2 (3.44 × 10−2) −	3.1380 × 10−2 (6.87 × 10−4)
MW5	2	15	1.2184 × 10−3 (1.44 × 10−3) −	3.6906 × 10−2 (7.46 × 10−2) −	1.0843 × 10−1 (7.38 × 10−2) −	1.4524 × 10−1 (8.43 × 10−2) −	1.6408 × 10−3 (1.88 × 10−3) −	3.8904 × 10−2 (7.42 × 10−2) −	3.7416 × 10−3 (5.76 × 10−3) −	4.6762 × 10−4 (1.43 × 10−4)
MW6	2	15	1.1606 × 10−2 (9.29 × 10−3) −	7.7872 × 10−2 (1.36 × 10−1) −	5.9216 × 10−1 (2.46 × 10−1) −	2.9330 × 10−1 (2.48 × 10−1) −	1.3406 × 10−2 (7.58 × 10−3) −	2.8856 × 10−2 (1.76 × 10−2) −	4.6464 × 10−2 (3.84 × 10−2) −	2.6789 × 10−3 (6.77 × 10−3)
MW7	2	15	2.4660 × 10−3 (5.13 × 10−4) ≈	2.4950 × 10−3 (1.35 × 10−3) −	6.4848 × 10−2 (5.32 × 10−2) −	8.7930 × 10−3 (3.04 × 10−3) −	2.3380 × 10−3 (5.50 × 10−4) +	5.9013 × 10−3 (1.52 × 10−3) −	2.5787 × 10−3 (3.56 × 10−4) ≈	2.3957 × 10−3 (1.75 × 10−4)
MW8	3	15	2.5003 × 10−2 (5.68 × 10−3) −	3.5904 × 10−2 (1.35 × 10−2) −	2.1834 × 10−1 (1.61 × 10−1) −	9.9667 × 10−2 (3.74 × 10−2) −	2.6727 × 10−2 (5.88 × 10−3) −	3.2696 × 10−2 (1.12 × 10−2) −	4.7679 × 10−2 (3.46 × 10−2) −	1.9549 × 10−2 (6.24 × 10−4)
MW9	2	15	4.2963 × 10−3 (9.92 × 10−4) −	9.2075 × 10−2 (2.03 × 10−1) −	1.8386 × 10−1 (3.12 × 10−1) −	9.8257 × 10−2 (1.78 × 10−1) −	6.0612 × 10−3 (1.01 × 10−2) −	7.8598 × 10−2 (1.62 × 10−1) −	2.0270 × 10−2 (7.13 × 10−2) −	3.1416 × 10−3 (3.32 × 10−4)
MW10	2	15	8.9651 × 10−2 (8.19 × 10−2) −	8.5892 × 10−2 (1.07 × 10−1) −	2.8026 × 10−1 (1.89 × 10−1) −	1.8764 × 10−1 (8.54 × 10−2) −	2.6201 × 10−2 (1.59 × 10−2) −	5.3440 × 10−2 (4.83 × 10−2) −	5.9470 × 10−2 (5.67 × 10−2) −	2.5048 × 10−3 (1.36 × 10−3)
MW11	2	15	1.5416 × 10−1 (2.37 × 10−1) −	2.7071 × 10−3 (1.67 × 10−4) ≈	9.4743 × 10−2 (1.33 × 10−1) −	2.5273 × 10−3 (1.47 × 10−4) +	1.9413 × 10−2 (1.13 × 10−2) −	6.1188 × 10−3 (1.99 × 10−3) −	3.2063 × 10−3 (1.92 × 10−4) −	2.6732 × 10−3 (9.73 × 10−5)
MW12	2	15	2.8967 × 10−1 (2.39 × 10−1) −	3.0585 × 10−3 (2.99 × 10−4) −	1.4341 × 10−2 (1.49 × 10−3) −	2.9910 × 10−3 (2.07 × 10−4) −	3.6519 × 10−3 (3.03 × 10−4) −	1.5711 × 10−1 (2.88 × 10−1) −	3.6537 × 10−3 (3.25 × 10−4) −	2.8454 × 10−3 (1.04 × 10−4)
MW13	2	15	8.7971 × 10−2 (6.91 × 10−2) −	8.0434 × 10−2 (5.40 × 10−2) −	7.3176 × 10−1 (5.86 × 10−1) −	1.2047 × 10−1 (7.05 × 10−2) −	2.7934 × 10−2 (1.30 × 10−2) −	6.3528 × 10−2 (4.36 × 10−2) −	5.4390 × 10−2 (6.64 × 10−2) −	1.7004 × 10−2 (4.63 × 10−2)
MW14	3	15	6.7516 × 10−1 (2.91 × 10−1) −	6.5065 × 10−2 (2.94 × 10−3) ≈	5.8381 × 10−1 (3.23 × 10−1) −	1.0256 × 100 (1.71 × 10−1) −	9.1428 × 10−2 (1.28 × 10−1) −	5.6201 × 10−1 (2.34 × 10−1) −	1.5499 × 10−1 (2.03 × 10−1) −	6.3681 × 10−2 (1.68 × 10−3)
+/−/≈			0/13/1	1/9/4	0/14/0	1/13/0	2/12/0	0/14/0	1/12/1

Table 2. Statistical analysis (mean ± standard deviation) of HV values for eight algorithms evaluated on MW benchmark problems.

Problem	M	D	BiCo	TSTI	POCEA	IMTCMO	TPCMaO	MTCMO	CMOEMT	CT-FTREA
MW1	2	15	3.1698 × 10−1 (1.16 × 10−1) −	4.7786 × 10−1 (2.70 × 10−2) ≈	4.6755 × 10−1 (7.91 × 10−3) −	4.8936 × 10−1 (4.63 × 10−4) −	4.8506 × 10−1 (1.21 × 10−2) −	3.7576 × 10−1 (9.36 × 10−2) −	4.8937 × 10−1 (7.55 × 10−4) −	4.8997 × 10−1 (4.42 × 10−5)
MW2	2	15	5.5857 × 10−1 (1.19 × 10−2) −	5.5993 × 10−1 (1.14 × 10−2) −	3.7057 × 10−1 (1.09 × 10−1) −	4.7248 × 10−1 (3.86 × 10−2) −	5.5321 × 10−1 (1.10 × 10−2) −	5.5334 × 10−1 (1.59 × 10−2) −	5.4138 × 10−1 (2.83 × 10−2) −	5.7995 × 10−1 (5.38 × 10−3)
MW3	2	15	5.1675 × 10−1 (1.39 × 10−2) −	5.3499 × 10−1 (2.28 × 10−2) ≈	4.9914 × 10−1 (1.64 × 10−2) −	5.2129 × 10−1 (1.15 × 10−2) −	5.4466 × 10−1 (5.87 × 10−4) +	5.3711 × 10−1 (2.82 × 10−3) −	5.4393 × 10−1 (7.39 × 10−4) +	5.4382 × 10−1 (2.27 × 10−4)
MW4	3	15	8.0085 × 10−1 (2.42 × 10−2) −	8.4104 × 10−1 (9.51 × 10−4) +	8.1488 × 10−1 (1.47 × 10−2) −	6.0023 × 10−1 (8.73 × 10−2) −	8.1537 × 10−1 (6.84 × 10−2) −	7.8824 × 10−1 (4.75 × 10−2) −	8.1598 × 10−1 (4.73 × 10−2) −	8.3936 × 10−1 (7.82 × 10−4)
MW5	2	15	3.2390 × 10−1 (9.41 × 10−4) −	2.9707 × 10−1 (5.45 × 10−2) −	2.2110 × 10−1 (5.50 × 10−2) −	1.5910 × 10−1 (7.52 × 10−2) −	3.2368 × 10−1 (1.18 × 10−3) −	2.8875 × 10−1 (5.11 × 10−2) −	3.2236 × 10−1 (3.99 × 10−3) −	3.2441 × 10−1 (9.38 × 10−5)
MW6	2	15	3.1407 × 10−1 (1.27 × 10−2) −	2.8392 × 10−1 (4.31 × 10−2) −	7.7752 × 10−2 (7.24 × 10−2) −	1.7663 × 10−1 (6.78 × 10−2) −	3.1070 × 10−1 (1.02 × 10−2) −	2.8848 × 10−1 (2.69 × 10−2) −	2.7597 × 10−1 (3.52 × 10−2) −	3.2645 × 10−1 (9.25 × 10−3)
MW7	2	15	4.1120 × 10−1 (1.18 × 10−3) ≈	4.1150 × 10−1 (2.06 × 10−3) +	3.4950 × 10−1 (3.43 × 10−2) −	3.9881 × 10−1 (5.82 × 10−3) −	4.1160 × 10−1 (8.39 × 10−4) +	4.0452 × 10−1 (3.01 × 10−3) −	4.1153 × 10−1 (7.63 × 10−4) +	4.1106 × 10−1 (4.73 × 10−4)
MW8	3	15	5.4198 × 10−1 (1.06 × 10−2) −	5.2150 × 10−1 (2.43 × 10−2) −	2.7958 × 10−1 (1.41 × 10−1) −	4.0302 × 10−1 (6.12 × 10−2) −	5.3131 × 10−1 (1.06 × 10−2) −	5.2628 × 10−1 (1.92 × 10−2) −	5.0076 × 10−1 (6.10 × 10−2) −	5.5122 × 10−1 (1.06 × 10−3)
MW9	2	15	3.9386 × 10−1 (2.75 × 10−3) −	3.3494 × 10−1 (1.30 × 10−1) −	2.6745 × 10−1 (1.51 × 10−1) −	3.1337 × 10−1 (8.16 × 10−2) −	3.9335 × 10−1 (1.71 × 10−2) ≈	3.2907 × 10−1 (1.06 × 10−1) −	3.7890 × 10−1 (7.32 × 10−2) ≈	3.9722 × 10−1 (2.09 × 10−3)
MW10	2	15	3.8009 × 10−1 (5.25 × 10−2) −	3.8393 × 10−1 (6.56 × 10−2) −	2.6164 × 10−1 (1.09 × 10−1) −	3.1779 × 10−1 (5.30 × 10−2) −	4.2520 × 10−1 (1.47 × 10−2) −	4.0572 × 10−1 (3.65 × 10−2) −	4.0262 × 10−1 (4.18 × 10−2) −	4.5422 × 10−1 (2.49 × 10−3)
MW11	2	15	3.8859 × 10−1 (7.55 × 10−2) −	4.4752 × 10−1 (3.99 × 10−4) ≈	4.0484 × 10−1 (4.21 × 10−2) −	4.4768 × 10−1 (3.17 × 10−4) +	4.3419 × 10−1 (7.17 × 10−3) −	4.4250 × 10−1 (2.55 × 10−3) −	4.4739 × 10−1 (2.31 × 10−4) ≈	4.4745 × 10−1 (2.12 × 10−4)
MW12	2	15	3.0440 × 10−1 (1.92 × 10−1) −	6.0470 × 10−1 (4.59 × 10−4) ≈	5.8680 × 10−1 (3.15 × 10−3) −	6.0450 × 10−1 (4.51 × 10−4) −	6.0364 × 10−1 (5.77 × 10−4) −	4.6646 × 10−1 (2.38 × 10−1) −	6.0361 × 10−1 (4.22 × 10−4) −	6.0487 × 10−1 (2.41 × 10−4)
MW13	2	15	4.1870 × 10−1 (3.24 × 10−2) −	4.2176 × 10−1 (3.06 × 10−2) −	1.9934 × 10−1 (1.23 × 10−1) −	3.9714 × 10−1 (3.91 × 10−2) −	4.5431 × 10−1 (1.04 × 10−2) −	4.3019 × 10−1 (2.52 × 10−2) −	4.3970 × 10−1 (3.89 × 10−2) −	4.6850 × 10−1 (2.85 × 10−2)
MW14	3	15	1.7742 × 10−1 (1.31 × 10−1) −	4.7297 × 10−1 (1.88 × 10−3) +	2.0625 × 10−1 (1.40 × 10−1) −	7.1427 × 10−2 (4.03 × 10−2) −	4.5049 × 10−1 (6.55 × 10−2) −	2.1964 × 10−1 (9.78 × 10−2) −	4.2558 × 10−1 (1.04 × 10−1) −	4.7040 × 10−1 (1.42 × 10−3)
+/−/≈			0/13/1	3/7/4	0/14/0	1/13/0	2/11/1	0/14/0	2/10/2

Furthermore, Figure 3 illustrates the solution distributions obtained on the MW6 problem, where the gray region represents the feasible space, the gray curve denotes the CPF, and the blue points correspond to feasible solutions. It can be observed that the solutions generated by CT-FTREA are distributed continuously and uniformly along the feasible boundary. In contrast, several competing algorithms, such as TSTI, POCEA, IMTCMO, and MTCMO, exhibit noticeable gaps or deviations in certain regions, reflecting limitations in their constraint-handling and search stability.

Figure 3. Final solution distributions of eight CMOEAs on the 2-objective MW12 problem.

In summary, the combined results from Table 1 and Table 2, and Figure 3 confirm that, through the synergistic integration of the constraint-tightening strategy and the feasible-trajectory guidance mechanism, CT-FTREA achieves consistently reliable performance in terms of convergence, solution distribution quality, and algorithmic stability. These advantages make it particularly suitable for tackling high-dimensional and complex constrained multi-objective optimization problems.

4.4.2. Comparative Analysis of LIRCMOP Problems

Table 3 and Table 4 summarize the IGD⁺ and HV statistics of eight algorithms on the LIRCMOP benchmark suite. From the IGD⁺ results, CT-FTREA demonstrates superior performance on most test problems. In particular, it achieves the smallest IGD⁺ values on LIRCMOP7–LIRCMOP9, LIRCMOP12, and LIRCMOP14, indicating strong approximation ability to the constrained Pareto front (CPF) across different objective dimensions. A comprehensive comparison in Table 3 shows that CT-FTREA outperforms BiCo, TSTI, POCEA, IMTCMO, TPCMaO, MTCMO, and CMOEMT on 12, 10, 13, 7, 7, 13, and 10 test problems, respectively, demonstrating stable convergence advantages across diverse constraint characteristics and superior overall performance. Regarding the HV metric (Table 4), CT-FTREA also exhibits clear advantages. The other algorithms are inferior on 12, 11, 12, 6, 4, 12, and 9 test problems, respectively, with CT-FTREA maintaining leading HV values, particularly on LIRCMOP7–LIRCMOP10 and the three-objective LIRCMOP14. This indicates that CT-FTREA not only improves the convergence of the solution set but also effectively preserves solution diversity and coverage in the objective space. Notably, even for problems with complex feasible regions (e.g., LIRCMOP5 and LIRCMOP6), CT-FTREA maintains relatively stable HV values, demonstrating its adaptability under complex constraints.

Table 3. Statistical analysis (mean ± standard deviation) of IGD⁺ values for eight algorithms evaluated on LIRCMOP benchmark problems.

Problem	M	D	BiCo	TSTI	POCEA	IMTCMO	TPCMaO	MTCMO	CMOEMT	CT-FTREA
LIRCMOP1	2	30	1.8670 × 10−1 (2.11 × 10−2) −	1.7957 × 10−1 (1.28 × 10−2) −	2.3881 × 10−1 (3.86 × 10−2) −	6.0818 × 10−2 (6.48 × 10−2) +	1.5749 × 10−1 (3.84 × 10−2) ≈	2.3169 × 10−1 (2.20 × 10−2) −	2.7753 × 10−1 (9.38 × 10−3) −	1.4620 × 10−1 (6.80 × 10−2)
LIRCMOP2	2	30	1.1600 × 10−1 (1.49 × 10−2) −	1.1746 × 10−1 (1.43 × 10−2) −	1.6806 × 10−1 (3.73 × 10−2) −	3.9469 × 10−2 (4.64 × 10−2) +	1.9903 × 10−1 (2.76 × 10−3) −	1.4958 × 10−1 (2.18 × 10−2) −	1.7226 × 10−2 (6.01 × 10−3) +	8.6656 × 10−2 (4.79 × 10−2)
LIRCMOP3	2	30	1.8846 × 10−1 (1.94 × 10−2) ≈	1.9087 × 10−1 (1.82 × 10−2) ≈	2.7158 × 10−1 (3.88 × 10−2) −	4.4862 × 10−2 (6.88 × 10−2) +	2.7809 × 10−1 (3.02 × 10−3) −	2.2604 × 10−1 (1.88 × 10−2) −	2.3588 × 10−2 (1.05 × 10−2) +	1.6310 × 10−1 (6.50 × 10−2)
LIRCMOP4	2	30	1.3100 × 10−1 (9.35 × 10−3) ≈	1.3492 × 10−1 (1.82 × 10−2) ≈	2.0929 × 10−1 (2.21 × 10−2) −	9.6419 × 10−2 (9.45 × 10−2) ≈	1.5103 × 10−1 (4.99 × 10−2) ≈	1.5866 × 10−1 (2.40 × 10−2) ≈	1.3176 × 10−1 (7.91 × 10−2) ≈	1.3963 × 10−1 (6.99 × 10−2)
LIRCMOP5	2	30	1.2219 × 100 (6.21 × 10−3) −	9.2591 × 10−1 (4.54 × 10−1) ≈	1.1148 × 100 (7.81 × 10−1) −	4.3083 × 100 (6.17 × 100) −	1.9187 × 10−1 (5.29 × 10−2) +	1.5826 × 100 (5.87 × 10−1) −	1.2602 × 100 (4.47 × 10−2) −	9.6892 × 10−1 (4.82 × 10−1)
LIRCMOP6	2	30	1.3453 × 100 (1.65 × 10−4) −	1.0426 × 100 (4.75 × 10−1) +	1.0917 × 100 (7.54 × 10−1) ≈	6.6141 × 100 (8.10 × 100) −	1.0423e+1 (8.45 × 100) −	1.4557 × 100 (3.39 × 10−1) −	1.3559 × 100 (9.48 × 10−3) −	1.1952 × 100 (3.68 × 10−1)
LIRCMOP7	2	30	5.0929 × 10−1 (6.95 × 10−1) −	1.1277 × 10−1 (2.25 × 10−2) −	1.4846 × 100 (9.50 × 10−1) −	2.8973 × 10−1 (6.05 × 10−1) ≈	7.4561 × 10−2 (1.83 × 10−2) ≈	1.5316 × 100 (4.67 × 10−1) −	9.7343 × 10−1 (7.41 × 10−1) −	6.5865 × 10−2 (3.80 × 10−2)
LIRCMOP8	2	30	1.0137 × 100 (7.58 × 10−1) −	3.4531 × 10−1 (4.58 × 10−1) −	9.7828 × 10−1 (7.39 × 10−1) −	4.3119 × 10−1 (7.43 × 10−1) −	1.1363 × 10−1 (3.53 × 10−2) −	1.6829 × 100 (7.54 × 10−4) −	9.5425 × 10−1 (6.95 × 10−1) −	7.6235 × 10−2 (6.91 × 10−2)
LIRCMOP9	2	30	9.3201 × 10−1 (1.08 × 10−1) −	3.0290 × 10−1 (7.93 × 10−2) −	1.1508 × 100 (1.65 × 10−1) −	3.0179 × 10−1 (1.41 × 10−1) −	3.8550 × 10−1 (1.06 × 10−1) −	1.0578 × 100 (5.06 × 10−2) −	4.3332 × 10−1 (4.04 × 10−1) −	2.5030 × 10−1 (1.27 × 10−1)
LIRCMOP10	2	30	9.2397 × 10−1 (8.68 × 10−2) −	5.2893 × 10−1 (2.54 × 10−1) −	1.0726 × 100 (1.47 × 10−1) −	2.1358 × 10−1 (3.17 × 10−1) ≈	1.6524 × 10−1 (5.28 × 10−2) ≈	1.0835 × 100 (4.96 × 10−2) −	1.0103 × 100 (9.56 × 10−2) −	1.8192 × 10−1 (1.39 × 10−1)
LIRCMOP11	2	30	6.7200 × 10−1 (2.28 × 10−1) −	5.5805 × 10−1 (1.76 × 10−1) −	1.1024 × 100 (1.41 × 10−1) −	5.3270 × 100 (4.05 × 100) −	6.5651 × 10−2 (4.39 × 10−2) ≈	9.9642 × 10−1 (8.04 × 10−2) −	9.8627 × 10−1 (1.34 × 10−1) −	1.1680 × 10−1 (1.29 × 10−1)
LIRCMOP12	2	30	5.9029 × 10−1 (2.49 × 10−1) −	2.6771 × 10−1 (8.08 × 10−2) −	9.7984 × 10−1 (2.52 × 10−1) −	2.2443 × 10−1 (1.32 × 10−1) ≈	3.5272 × 100 (6.19 × 100) −	9.5200 × 10−1 (1.20 × 10−1) −	9.6728 × 10−1 (1.81 × 10−1) −	1.3645 × 10−1 (7.62 × 10−2)
LIRCMOP13	3	30	1.3171 × 100 (1.83 × 10−3) −	1.1897 × 100 (3.89 × 10−1) −	1.3080 × 100 (2.39 × 10−1) −	1.3137 × 100 (7.73 × 10−3) −	4.1369 × 10−2 (1.13 × 10−3) +	1.3186 × 100 (2.76 × 10−3) −	6.0718 × 10−1 (2.19 × 100) ≈	6.6042 × 10−2 (2.32 × 10−3)
LIRCMOP14	3	30	1.2738 × 100 (2.05 × 10−3) −	1.2704 × 100 (1.76 × 10−3) −	1.3289 × 100 (1.93 × 10−1) −	1.2696 × 100 (6.13 × 10−3) −	6.4957 × 10−2 (6.47 × 10−3) −	1.2753 × 100 (2.19 × 10−3) −	5.1385 × 10−2 (1.94 × 10−3) −	4.9940 × 10−2 (8.75 × 10−4)
+/−/≈			0/12/2	1/10/3	0/13/1	3/7/4	2/7/5	0/13/1	2/10/2

Table 4. Statistical analysis (mean ± standard deviation) of HV values for eight algorithms evaluated on LIRCMOP benchmark problems.

Problem	M	D	BiCo	TSTI	POCEA	IMTCMO	TPCMaO	MTCMO	CMOEMT	CT-FTREA
LIRCMOP1	2	30	1.3937 × 10−1 (9.00 × 10−3) −	1.4021 × 10−1 (7.42 × 10−3) −	1.1605 × 10−1 (1.65 × 10−2) −	2.0341 × 10−1 (3.63 × 10−2) +	1.5025 × 10−1 (2.06 × 10−2) ≈	1.2036 × 10−1 (9.81 × 10−3) −	1.0125 × 10−1 (3.78 × 10−3) −	1.5373 × 10−1 (3.20 × 10−2)
LIRCMOP2	2	30	2.5706 × 10−1 (1.25 × 10−2) −	2.5862 × 10−1 (8.97 × 10−3) −	2.2766 × 10−1 (2.32 × 10−2) −	3.3015 × 10−1 (4.12 × 10−2) +	2.0796 × 10−1 (3.83 × 10−3) −	2.3658 × 10−1 (1.45 × 10−2) −	3.5182 × 10−1 (4.11 × 10−3) +	2.9336 × 10−1 (3.33 × 10−2)
LIRCMOP3	2	30	1.2596 × 10−1 (8.39 × 10−3) −	1.2362 × 10−1 (9.82 × 10−3) −	9.8029 × 10−2 (1.06 × 10−2) −	1.8524 × 10−1 (3.01 × 10−2) +	8.7805 × 10−2 (3.63 × 10−3) −	1.0866 × 10−1 (8.85 × 10−3) −	1.9249 × 10−1 (7.67 × 10−3) +	1.3734 × 10−1 (1.93 × 10−2)
LIRCMOP4	2	30	2.2302 × 10−1 (7.89 × 10−3) ≈	2.2253 × 10−1 (1.24 × 10−2) ≈	1.8040 × 10−1 (1.30 × 10−2) −	2.5354 × 10−1 (5.99 × 10−2) ≈	2.1821 × 10−1 (3.39 × 10−2) ≈	2.0745 × 10−1 (1.44 × 10−2) ≈	2.2766 × 10−1 (5.18 × 10−2) ≈	2.2501 × 10−1 (3.94 × 10−2)
LIRCMOP5	2	30	0.0000 × 100 (0.00 × 100) −	4.3254 × 10−2 (6.83 × 10−2) ≈	4.6659 × 10−2 (6.66 × 10−2) ≈	0.0000 × 100 (0.00 × 100) −	1.7028 × 10−1 (2.32 × 10−2) +	0.0000 × 100 (0.00 × 100) −	0.0000 × 100 (0.00 × 100) −	5.5532 × 10−2 (1.14 × 10−1)
LIRCMOP6	2	30	0.0000 × 100 (0.00 × 100) ≈	2.7718 × 10−2 (4.35 × 10−2) ≈	3.4132 × 10−2 (4.29 × 10−2) ≈	2.8501 × 10−2 (6.55 × 10−2) ≈	4.0440 × 10−2 (6.38 × 10−2) ≈	0.0000 × 100 (0.00 × 100) ≈	0.0000 × 100 (0.00 × 100) ≈	1.2569 × 10−2 (3.17 × 10−2)
LIRCMOP7	2	30	1.8117 × 10−1 (1.08 × 10−1) −	2.4395 × 10−1 (9.12 × 10−3) −	5.7636 × 10−2 (1.02 × 10−1) −	2.3539 × 10−1 (1.05 × 10−1) ≈	2.5829 × 10−1 (7.43 × 10−3) ≈	2.2437 × 10−2 (6.91 × 10−2) −	1.0283 × 10−1 (1.07 × 10−1) −	2.6394 × 10−1 (1.78 × 10−2)
LIRCMOP8	2	30	9.9586 × 10−2 (1.13 × 10−1) −	2.0098 × 10−1 (6.89 × 10−2) −	1.0167 × 10−1 (1.05 × 10−1) −	2.1819 × 10−1 (1.30 × 10−1) −	2.4695 × 10−1 (1.02 × 10−2) ≈	0.0000 × 100 (0.00 × 100) −	9.9297 × 10−2 (9.45 × 10−2) −	2.6393 × 10−1 (2.61 × 10−2)
LIRCMOP9	2	30	1.3088 × 10−1 (4.68 × 10−2) −	3.5465 × 10−1 (4.14 × 10−2) −	7.2623 × 10−2 (5.35 × 10−2) −	3.6292 × 10−1 (9.08 × 10−2) −	3.5668 × 10−1 (5.50 × 10−2) −	8.8436 × 10−2 (2.02 × 10−2) −	3.4249 × 10−1 (1.53 × 10−1) −	4.2192 × 10−1 (7.23 × 10−2)
LIRCMOP10	2	30	7.8375 × 10−2 (3.60 × 10−2) −	3.3427 × 10−1 (1.78 × 10−1) −	5.9589 × 10−2 (2.73 × 10−2) −	5.4951 × 10−1 (2.39 × 10−1) ≈	5.9071 × 10−1 (3.71 × 10−2) ≈	5.0698 × 10−2 (2.29 × 10−3) −	6.7502 × 10−2 (3.80 × 10−2) −	5.8383 × 10−1 (1.14 × 10−1)
LIRCMOP11	2	30	2.7374 × 10−1 (1.05 × 10−1) −	3.3182 × 10−1 (8.74 × 10−2) −	1.1805 × 10−1 (5.18 × 10−2) −	2.0567 × 10−1 (2.97 × 10−1) −	6.5200 × 10−1 (3.60 × 10−2) ≈	1.5729 × 10−1 (2.57 × 10−2) −	1.5574 × 10−1 (5.05 × 10−2) −	6.0145 × 10−1 (1.07 × 10−1)
LIRCMOP12	2	30	3.4543 × 10−1 (1.26 × 10−1) −	4.4345 × 10−1 (5.44 × 10−2) −	2.4214 × 10−1 (9.50 × 10−2) −	4.7479 × 10−1 (9.20 × 10−2) ≈	3.0848 × 10−1 (2.33 × 10−1) −	2.0156 × 10−1 (6.12 × 10−2) −	2.0031 × 10−1 (7.50 × 10−2) −	5.3686 × 10−1 (5.02 × 10−2)
LIRCMOP13	3	30	1.0027 × 10−4 (9.83 × 10−5) −	5.4642 × 10−2 (1.68 × 10−1) −	1.7652 × 10−2 (4.66 × 10−2) −	1.0481 × 10−4 (1.39 × 10−4) −	5.6108 × 10−1 (9.74 × 10−4) +	1.6081 × 10−4 (1.45 × 10−4) −	4.5839 × 10−1 (1.35 × 10−1) ≈	5.3281 × 10−1 (2.48 × 10−3)
LIRCMOP14	3	30	4.3442 × 10−4 (2.88 × 10−4) −	5.1083 × 10−4 (3.49 × 10−4) −	1.0184 × 10−2 (4.54 × 10−2) −	4.6347 × 10−4 (3.35 × 10−4) −	5.4987 × 10−1 (1.81 × 10−3) ≈	5.1252 × 10−4 (3.14 × 10−4) −	5.4830 × 10−1 (2.20 × 10−3) −	5.5028 × 10−1 (8.02 × 10−4)
+/−/≈			0/12/2	0/11/3	0/12/2	3/6/5	2/4/8	0/12/2	2/9/3

Figure 4 shows the final solution distributions on the LIRCMOP14 problem. The red circles represent feasible solutions, while the gray curve denotes the constrained Pareto front (CPF). It can be observed that CT-FTREA produces solutions that are continuous and well-distributed along the constrained Pareto front, whereas some comparison algorithms exhibit sparse or incomplete coverage, reflecting limitations in constraint handling and search stability.

Figure 4. Final solution distributions of eight CMOEAs on the 3-objective LIRCMOP14 problem.

Overall, considering IGD⁺, HV, and solution distribution, CT-FTREA consistently demonstrates advantages in convergence accuracy, solution quality, and search stability, making it highly robust and effective for complex, high-dimensional constrained multi-objective optimization problems.

4.4.3. Comparative Analysis of Real-World CMOP

Based on the results in Table 5, the performance of eight algorithms on five real-world engineering problems—Vibrating Platform (VBP), Two-Bar Truss Design (TBTD), Welded Beam Design Problem (WBDP), Disc Brake Design Problem (DBDP), and Gear Train Design Problem (GTDP)—can be summarized. Since the true Pareto front is unknown, the HV indicator is commonly used to evaluate algorithm performance, where higher HV values indicate broader coverage and higher-quality solution sets. Overall, CT-FTREA achieves the highest or near-highest HV values across these problems, generating solution sets that are generally stable and reasonably distributed, although differences among the algorithms are not always substantial.

Table 5. Statistical analysis (mean ± standard deviation) of HV values for eight algorithms evaluated on real-world problems.

Problem	M	D	BiCo	TSTI	POCEA	IMTCMO	TPCMaO	MTCMO	CMOEMT	CT-FTREA
VBP	2	5	3.7500 × 10−1 (2.80 × 10−2) −	2.7506 × 10−1 (1.52 × 10−1) −	1.4469 × 10−1 (9.11 × 10−2) −	3.9292 × 10−1 (1.49 × 10−5) −	3.3757 × 10−1 (7.23 × 10−2) −	3.9286 × 10−1 (1.28 × 10−4) −	3.9290 × 10−1 (3.93 × 10−5) −	3.9300 × 10−1 (6.63 × 10−6)
TBTD	2	3	8.9978 × 10−1 (5.37 × 10−4) −	8.9927 × 10−1 (4.62 × 10−4) −	2.3729 × 10−1 (2.09 × 10−1) −	8.9936 × 10−1 (5.87 × 10−4) −	8.9847 × 10−1 (6.84 × 10−4) −	8.9953 × 10−1 (4.52 × 10−4) −	8.9840 × 10−1 (9.34 × 10−4) −	9.0245 × 10−1 (7.98 × 10−5)
WBDP	2	4	8.5888 × 10−1 (2.40 × 10−3) −	8.5795 × 10−1 (4.49 × 10−3) −	4.8728 × 10−3 (2.12 × 10−2) −	8.6005 × 10−1 (6.19 × 10−4) −	8.6037 × 10−1 (1.68 × 10−3) −	8.5990 × 10−1 (1.33 × 10−3) −	8.5666 × 10−1 (9.96 × 10−4) −	8.6268 × 10−1 (7.48 × 10−4)
DBDP	2	4	4.3496 × 10−1 (1.11 × 10−4) −	4.3446 × 10−1 (5.16 × 10−4) −	4.2345 × 10−1 (1.96 × 10−3) −	4.3493 × 10−1 (1.41 × 10−4) −	3.9561 × 10−1 (1.54 × 10−2) −	4.3493 × 10−1 (9.68 × 10−5) −	4.3449 × 10−1 (1.07 × 10−4) −	4.3519 × 10−1 (6.73 × 10−5)
GTDP	2	4	4.8459 × 10−1 (4.08 × 10−5) −	4.8456 × 10−1 (6.52 × 10−5) −	4.8076 × 10−1 (1.03 × 10−3) −	4.8458 × 10−1 (4.67 × 10−5) −	4.8298 × 10−1 (1.04 × 10−3) −	4.8455 × 10−1 (3.53 × 10−5) −	4.8462 × 10−1 (3.50 × 10−5) −	4.8468 × 10−1 (3.97 × 10−5)
+/−/≈			0/5/0	0/5/0	0/5/0	0/5/0	0/5/0	0/5/0	0/5/0

Compared with the other seven algorithms (BiCo, TSTI, POCEA, IMTCMO, TPCMaO, MTCMO, and CMOEMT), CT-FTREA demonstrates relatively stable performance across all tested problems. Some algorithms exhibit lower HV values or larger fluctuations in certain cases, indicating limited coverage or less consistent convergence. In contrast, CT-FTREA generally maintains high HV values with smaller standard deviations, reflecting a more balanced performance. Overall, CT-FTREA can produce reliable Pareto solution sets for a variety of real-world constrained multi-objective optimization problems, providing practical guidance for engineering design. In terms of search capability, solution quality, and algorithm stability, CT-FTREA shows a well-balanced performance compared with other methods, highlighting its potential for practical applications.

4.4.4. Ablation Study on CT-FTREA

To assess the contribution of the core modules of CT-FTREA to the algorithm’s performance, three ablation experiments were designed in this paper:

(1)

CT-FTREA1: The two-stage optimization framework (feasible-region-guided stage + CPF-focused refinement stage) is removed, and the entire optimization process is simplified into a single-stage search. This experiment aims to verify the effect of the two-stage design on convergence and the quality of the solution set distribution.

(2)

CT-FTREA2: The adaptive constraint boundary tightening strategy is removed and replaced with a fixed constraint boundary. This experiment examines the contribution of the strategy in guiding the population into the feasible region, increasing the proportion of feasible solutions, and improving the approximation of the CPF.

(3)

CT-FTREA3: The elite environmental selection strategy is removed, retaining only the basic selection operator. This experiment analyzes the impact of elite environmental selection on the stability of the solution set, convergence, and the approximation ability of the Pareto front.

These three ablation algorithms were tested on the MW benchmark problem, and the effects of each module were analyzed using the IGD⁺ and HV metrics. According to the results of the ablation experiments shown in Table 6 and Table 7, it is evident that all core components of CT-FTREA play an important role in its overall performance. In particular, the two-stage optimization framework and the adaptive constraint boundary tightening strategy are crucial for handling narrow and complex constrained problems, ensuring faster convergence and better solution set distribution. The elite environmental selection strategy further enhances the stability of the search process and improves the approximation accuracy of the constrained Pareto front. Removing any of these components significantly reduces the overall performance of the algorithm, highlighting their importance in the proposed algorithm.

Table 6. Statistical IGD⁺ (mean ± standard deviation) validation experiments of CT-FTREA and its three variants on the DASCMOP test problems.

Problem	M	D	CT-FTREA1	CT-FTREA2	CT-FTREA3	CT-FTREA
MW1	2	15	9.1918 × 10−3 (8.53 × 10−3) −	2.3776 × 10−2 (9.59 × 10−2) −	5.2980 × 10−3 (9.99 × 10−3) ≈	1.2127 × 10−3 (3.16 × 10−5)
MW2	2	15	3.6269 × 10−2 (7.90 × 10−2) ≈	1.1090 × 10−1 (9.06 × 10−2) −	6.2182 × 10−3 (7.84 × 10−3) −	4.0504 × 10−3 (3.00 × 10−3)
MW3	2	15	3.8785 × 10−2 (1.10 × 10−1) −	4.9154 × 10−3 (4.66 × 10−3) ≈	5.0577 × 10−3 (4.63 × 10−3) ≈	3.3120 × 10−3 (1.42 × 10−4)
MW4	3	15	3.8892 × 10−2 (3.14 × 10−3) −	3.0213 × 10−2 (6.91 × 10−4) +	3.8245 × 10−2 (1.66 × 10−2) ≈	3.1380 × 10−2 (6.87 × 10−4)
MW5	2	15	5.3332 × 10−3 (2.40 × 10−3) −	1.7375 × 10−1 (2.32 × 10−1) −	4.5009 × 10−3 (1.12 × 10−2) −	4.6762 × 10−4 (1.43 × 10−4)
MW6	2	15	4.7823 × 10−2 (1.43 × 10−1) ≈	4.0772 × 10−1 (3.89 × 10−1) −	6.3827 × 10−3 (1.12 × 10−2) ≈	2.6789 × 10−3 (6.77 × 10−3)
MW7	2	15	2.8950 × 10−2 (8.05 × 10−2) −	4.3766 × 10−2 (1.24 × 10−1) −	9.0960 × 10−3 (3.97 × 10−3) −	2.3957 × 10−3 (1.75 × 10−4)
MW8	3	15	3.1581 × 10−2 (3.56 × 10−2) ≈	1.3311 × 10−1 (1.41 × 10−1) −	4.6422 × 10−2 (8.20 × 10−3) −	1.9549 × 10−2 (6.24 × 10−4)
MW9	2	15	1.7458 × 10−2 (5.04 × 10−3) −	2.0056 × 10−2 (4.48 × 10−2) −	1.2036 × 10−2 (1.14 × 10−2) −	3.1416 × 10−3 (3.32 × 10−4)
MW10	2	15	4.4760 × 10−3 (4.43 × 10−3) ≈	2.1799 × 10−1 (1.64 × 10−1) −	1.8803 × 10−2 (4.15 × 10−2) ≈	2.5048 × 10−3 (1.36 × 10−3)
MW11	2	15	8.0518 × 10−2 (1.89 × 10−1) −	4.7571 × 10−3 (3.43 × 10−3) −	4.2464 × 10−3 (2.42 × 10−3) −	2.6732 × 10−3 (9.73 × 10−5)
MW12	2	15	4.1439 × 10−3 (1.71 × 10−3) −	3.1721 × 10−3 (1.36 × 10−4) −	6.4323 × 10−3 (6.76 × 10−3) −	2.8454 × 10−3 (1.04 × 10−4)
MW13	2	15	1.0107 × 10−1 (2.20 × 10−1) −	3.5643 × 10−1 (3.32 × 10−1) −	3.6372 × 10−2 (7.06 × 10−2) ≈	1.7004 × 10−2 (4.63 × 10−2)
MW14	3	15	2.1182 × 10−1 (3.31 × 10−1) −	1.4217 × 10−1 (2.06 × 10−1) ≈	1.5545 × 10−1 (2.83 × 10−1) ≈	6.3681 × 10−2 (1.68 × 10−3)
+/−/≈			0/10/4	1/11/2	0/7/7

Table 7. Statistical HV (mean ± standard deviation) validation experiments of HMP-CE and its three variants on the DASCMOP test problems.

Problem	M	D	CT-FTREA1	CT-FTREA2	CT-FTREA3	CT-FTREA
MW1	2	15	4.7435 × 10−1 (1.38 × 10−2) −	4.6538 × 10−1 (1.05 × 10−1) −	4.8234 × 10−1 (1.87 × 10−2) ≈	4.8997 × 10−1 (4.42 × 10−5)
MW2	2	15	5.3441 × 10−1 (1.12 × 10−1) ≈	4.3879 × 10−1 (1.02 × 10−1) −	5.7617 × 10−1 (1.37 × 10−2) −	5.7995 × 10−1 (5.38 × 10−3)
MW3	2	15	5.0448 × 10−1 (1.16 × 10−1) −	5.4134 × 10−1 (7.35 × 10−3) ≈	5.4054 × 10−1 (8.66 × 10−3) ≈	5.4382 × 10−1 (2.27 × 10−4)
MW4	3	15	8.3126 × 10−1 (3.45 × 10−3) −	8.4068 × 10−1 (7.18 × 10−4) +	8.3115 × 10−1 (1.98 × 10−2) ≈	8.3936 × 10−1 (7.82 × 10−4)
MW5	2	15	3.2088 × 10−1 (1.72 × 10−3) −	2.1580 × 10−1 (1.20 × 10−1) −	3.2073 × 10−1 (1.05 × 10−2) −	3.2441 × 10−1 (9.38 × 10−5)
MW6	2	15	3.0792 × 10−1 (6.16 × 10−2) −	1.4942 × 10−1 (1.06 × 10−1) −	3.1986 × 10−1 (1.88 × 10−2) ≈	3.2645 × 10−1 (9.25 × 10−3)
MW7	2	15	3.8201 × 10−1 (8.85 × 10−2) ≈	3.8291 × 10−1 (7.80 × 10−2) −	3.9925 × 10−1 (7.24 × 10−3) −	4.1106 × 10−1 (4.73 × 10−4)
MW8	3	15	5.2701 × 10−1 (7.29 × 10−2) ≈	3.9600 × 10−1 (1.34 × 10−1) −	4.9382 × 10−1 (1.63 × 10−2) −	5.5122 × 10−1 (1.06 × 10−3)
MW9	2	15	3.7393 × 10−1 (5.35 × 10−3) −	3.7459 × 10−1 (3.92 × 10−2) −	3.8086 × 10−1 (1.40 × 10−2) −	3.9722 × 10−1 (2.09 × 10−3)
MW10	2	15	4.5085 × 10−1 (6.75 × 10−3) ≈	3.0416 × 10−1 (9.36 × 10−2) −	4.4002 × 10−1 (3.47 × 10−2) −	4.5422 × 10−1 (2.49 × 10−3)
MW11	2	15	4.1593 × 10−1 (7.58 × 10−2) −	4.4560 × 10−1 (2.48 × 10−3) −	4.4586 × 10−1 (2.32 × 10−3) −	4.4745 × 10−1 (2.12 × 10−4)
MW12	2	15	6.0203 × 10−1 (3.65 × 10−3) −	6.0404 × 10−1 (4.48 × 10−4) −	5.9793 × 10−1 (1.20 × 10−2) −	6.0487 × 10−1 (2.41 × 10−4)
MW13	2	15	4.1820 × 10−1 (1.17 × 10−1) −	2.8161 × 10−1 (1.29 × 10−1) −	4.5381 × 10−1 (4.50 × 10−2) −	4.6850 × 10−1 (2.85 × 10−2)
MW14	3	15	4.0631 × 10−1 (1.34 × 10−1) −	4.2976 × 10−1 (1.03 × 10−1) ≈	4.3050 × 10−1 (1.23 × 10−1) ≈	4.7040 × 10−1 (1.42 × 10−3)
+/−/≈			0/10/4	1/11/2	0/9/5

5. Conclusions

This paper presents a novel CT-FTREA designed to address CMOPs. The algorithm effectively tackles the challenges posed by complex constraints, enhancing both search efficiency and solution quality. By dividing the optimization process into two stages—the feasible-region-guided stage and the CPF-focused refinement stage—CT-FTREA adapts to the dynamics of constraint handling, ensuring better convergence and diversity in the solution set. In the first stage, the algorithm employs an adaptive constraint boundary tightening strategy, gradually guiding the population from infeasible regions into the feasible region. This approach, combined with a weighted fitness evaluation mechanism and an elite archive strategy, facilitates efficient exploration of the feasible Pareto front. In the second stage, CT-FTREA exchanges the roles of the population and the archive, refocusing on the fine-grained approximation of the constrained Pareto front. This exchange process, combined with an improved elite selection strategy and differential evolution operator, ensures a balanced exploration and exploitation of the solution space.

In summary, CT-FTREA demonstrates robust performance across benchmark tests and real-world applications, highlighting its potential in practical engineering design and multi-objective optimization. However, the algorithm has limitations, particularly when dealing with highly dynamic or large-scale problems, where computational complexity may become a challenge. Future work will focus on extending the algorithm to handle dynamic environments, improving scalability, and addressing computational challenges associated with large-dimensional, multi-constraint CMOPs.