A Dual-Stage and Dual-Population Algorithm Based on Chemical Reaction Optimization for Constrained Multi-Objective Optimization

Abstract

Constrained multi-objective optimization problems (CMOPs) require optimizing multiple conflicting objectives while satisfying complex constraints. These constraints generate infeasible regions that challenge traditional algorithms in balancing feasibility and Pareto frontier diversity. chemical reaction optimization (CRO) effectively balances global exploration and local exploitation through molecular collision reactions and energy management, thereby enhancing search efficiency. However, standard CRO variants often struggle with CMOPs due to the absence of specialized constraint-handling mechanisms. To address these challenges, this paper integrates the CRO collision reaction mechanism with an existing evolutionary computational framework to design a dual-stage and dual-population chemical reaction optimization (DDCRO) algorithm. This approach employs a staged optimization strategy, which divides population evolution into two phases. The first phase focuses on objective optimization to enhance population diversity, and the second prioritizes constraint satisfaction to accelerate convergence toward the constrained Pareto front. Furthermore, to leverage the infeasible solutions’ guiding potential during the search, DDCRO adopts a two-population strategy. At each stage, the main population tackles the original constrained problem, while the auxiliary population addresses the corresponding unconstrained version. A weak complementary mechanism facilitates information sharing between populations, which enhances search efficiency and algorithmic robustness. Comparative tests on multiple test suites reveal that DDCRO achieves optimal IGD/HV values in 53% of test problems. The proposed algorithm outperforms other state-of-the-art algorithms in both convergence and population diversity.

1. Introduction

Multi-objective optimization problems (MOPs) are widely used in scientific research and engineering practice and usually involve multiple conflicting objectives that need to be optimized [1]. As real-world challenges grow in complexity, the introduction of constraints has driven the evolution of these problems into constrained multi-objective optimization problems (CMOPs) [2]. For example, the design of aerospace structures needs to satisfy material strength and weight constraints, and power dispatch needs to follow the constraints of the safe operation of power grids [3,4]. The introduction of constraints in CMOPs significantly increases the complexity of the problem-solving. On the one hand, the introduction of constraints leads to the proliferation of infeasible solutions, exacerbates the search difficulty, and severely weakens the convergence efficiency of the population to the constrained Pareto front (CPF) due to the blockage of infeasible domains in the objective space. On the other hand, when the CPF shows a discontinuous distribution, the algorithm faces challenges in maintaining the balance between objective optimization and constraint satisfaction, which becomes the core research difficulty in the field of constrained multi-objective optimization.

To address the aforementioned challenges, researchers have developed a diverse array of optimization methodologies. These approaches can be categorized into four principal strategies based on their constraint-handling mechanisms: constrained dominance principle (CDP) [5], penalty function methods [6], $ϵ$-constraint techniques [7], and hybrid strategies [8]. The CDP strategy establishes a hierarchical decision-making framework for solution comparison [9]. Owing to its simplicity and operational ease, CDP has been widely implemented in algorithms such as CDP-NSGA-II [5], CMOEA/D [9], and SPEA2-CDP [10]. Penalty function approaches transform constrained problems into unconstrained counterparts by incorporating constraint violation (CV) metrics into the objective function through penalty terms. These methods introduce penalty factors as adjustable parameters, maintaining the original problem dimensionality while simplifying constraint handling [6]. In order to adapt to different types of constrained multi-objective optimization problems, some cutting-edge algorithms, such as PF-ICMOA [11] and ICMOEA/D [12], design the update mechanism of the penalty factor, so that the algorithm can adjust the priority of the constraints according to the convergence of the population. The $ϵ$-based constraint technique, first proposed by Takahama and Sakathe, aims to relax the constraints by thresholding $ϵ$. And the solution is considered feasible when the degree of constraint violation of an individual is less than $ϵ$. Thus, the technique is able to guide the population evolution by retaining some of the infeasible solutions and fully utilizing the valid information contained in these solutions [7]. The performance of this approach critically depends on $ϵ$ threshold selection, which prompts the development of various dynamic adjustment strategies by researchers such as Song [13], Ji [14], and Wang [15]. These advancements have enriched the algorithm design framework based on the $ϵ$-constraint technique. The hybrid approach mainly improves the performance of the algorithm in terms of population convergence and diversity by combining multiple evolutionary strategies or constraint processing mechanisms. The approach typically integrates mathematical optimization methods with search procedures. Notable examples include URCMO [8], which employs dynamic operator selection based on CPF/UPF relationships, and DDCEA [16], which implements phase-specific switching between genetic algorithm (GA) and differential evolution (DE) operators to enhance global exploration and local exploitation capabilities, respectively.

The CDP and penalty function strategies mentioned above can show their superiority in some test problems where the constraints are uniformly shaped and the distribution has obvious regularity. However, when the feasible domain is more complex, such as when the feasible domain is narrow, discrete, disjoint, etc., the algorithms using these two techniques often have difficulty in searching the global optimal feasible frontier. This is mainly attributed to the fact that the selection pressure of both techniques cannot be adjusted adaptively to cope with problems in complex infeasible domains. High selection pressure tends to promote rapid convergence of the population, but may lead to premature convergence, while low selection pressure is conducive to maintaining the diversity of the population, but may slow down the convergence speed. The existing $ϵ$-constraint-based techniques can dynamically adjust the selection pressure, but they fail to combine the adaptive mechanism of the constraint violation threshold with the problem characteristics, resulting in poor robustness and generalizability of the algorithms. Although the hybrid methods can balance the performance of the algorithms well, they generally have the problem of poor algorithm generalization and need to adopt different strategies for different types of CMOPs. Overall, how to effectively balance population convergence, diversity, and feasibility in CMOPs remains a research focus and difficulty in the field of constrained optimization. In recent years, researchers have proposed multi-stage optimization strategies [17,18,19], which better solve the balance problem between constrained and objective optimization through the choice of staged strategies. This strategy provides important enlightenment for the research of this paper.

Chemical reaction optimization (CRO) algorithms achieve global and local search within the solution space by simulating molecular collision reactions. These algorithms dynamically eliminate low-potential individuals based on the energy management criterion to significantly optimize the allocation of computational resources [20,21]. The flexibility and diversity inherent in its collision reaction mechanisms provide substantial opportunities for optimization enhancement. This endows it with greater potential to explore complex CMOP landscapes characterized by fragmented feasible regions. Meanwhile, it has been shown that the environment selection process and offspring generation process are pivotal in the design of optimization algorithms [22]. Although traditional frameworks such as differential evolution (DE) and genetic algorithm (GA) demonstrate strong performance in simple constrained optimization, their reliance on static penalty functions or fixed operators restricts adaptability to complex CMOPs characterized by irregular infeasible regions and rigidifies the mechanisms for offspring generation [23]. Consequently, integrating the distinctive advantages of CRO algorithms—particularly in offspring generation—with existing constrained optimization methods holds substantial research value and potential. Previous studies on CRO algorithms have demonstrated their effectiveness in solving unconstrained and some constrained optimization problems. However, existing CRO-based algorithms often lack specialized constraint-handling mechanisms and struggle to balance objective optimization with feasibility convergence in complex CMOPs. Specifically, they tend to have difficulty in navigating complex infeasible regions, resulting in premature convergence or the inability to find high-quality feasible solutions. Additionally, the single population structure in traditional CRO algorithms restricts exploration efficiency by limiting diversity maintenance and adaptive search direction adjustments.

Fortunately, the flexible reaction mechanisms inherent in CRO naturally align with the collaborative evolution principles, thereby paving the way for enhanced optimization performance. To break through this existing bottleneck, we consider the use of a dual-population synergy framework and accordingly propose a dual-stage and dual-population constrained multi-objective evolutionary algorithm (DDCRO) rooted in CRO principles. The proposed DDCRO integrates an enhanced CRO algorithm with a dual-stage optimization strategy. The core framework achieves the dynamic balance of objective optimization and constraint satisfaction through a staged strategy and a dual-population synergistic mechanism. Specifically, the main population focuses on the original constraint problem and adopts diversity collision operators and convergence collision operators in phases to strengthen the global exploration and local exploitation capabilities, respectively. Meanwhile, the auxiliary population generates bootstrap infeasible solutions by optimizing the unconstrained problem, and combines with the weak complementary mechanism to share the information of offspring with the main population to effectively break through the barrier of infeasible domains. Moreover, DDCRO refines the response mechanism of the traditional CRO algorithm by introducing a structure repair strategy for historical optimal solutions during decomposition and synthesis, thereby preserving high-quality solution information. It also monitors population convergence through a dynamic threshold, adaptively triggering stage transitions to ensure optimal performance. In the experimental section, DDCRO undergoes a comprehensive evaluation using mainstream complex constrained multi-objective test sets. The results demonstrate that DDCRO significantly surpasses existing state-of-the-art methods in terms of CPF diversity, convergence speed, and feasible solution coverage. This verifies its superiority and robustness in addressing problems characterized by discontinuous, narrow, and discrete feasible domains.

The main contributions of this paper are as follows:

• Innovative integration of CRO with a dual-stage and dual-population framework: Uniquely integrates the collision reaction mechanism of CRO with a tailored dual-stage and dual-population framework specifically designed for CMOPs. This fusion not only leverages CRO’s strengths in global and local search but also addresses its limitations in handling complex constraints and infeasible regions through a structured, phased approach.
• Enhanced molecular collision operators with structure repair strategy: Introduces novel molecular collision operators that incorporate a structure repair strategy to preserve high-quality solution structures during decomposition and synthesis reactions. This strategy prevents the loss of historically optimal structures throughout the reaction process, which enhances the algorithm’s search efficiency and stability.
• Dynamic stage transition mechanism based on population state: Implements a dynamic stage transition mechanism that switches from the first stage (objective optimization) to the second stage (constraint satisfaction) based on the actual state of the population. This mechanism ensures that the algorithm can adaptively respond to problem characteristics, which facilitates efficient convergence towards the CPF.
• Weak complementary mechanism for inter-population information exchange: Designs a weak complementary mechanism that promotes information exchange between the main and auxiliary populations without direct substitution of individuals. This mechanism harnesses the guiding potential of infeasible solutions while maintaining the feasibility and diversity of the main population, thereby enhancing the overall performance of the algorithm.

2. Related Work

2.1. Multi-Objective Evolutionary Algorithm Based on Chemical Reaction Optimization

Chemical reaction optimization is a meta-heuristic algorithm that simulates the dynamics of molecular collisions. It designs four basic chemical collision reactions: on-wall ineffective collision, decomposition reaction, inter-molecular ineffective collision, and synthesis reaction. Through the synergistic effect of the four types of reaction mechanisms, the global exploration and local development of the solution space are achieved [24]. Among them, uni-molecular reactions (on-wall ineffective collision/decomposition reaction) dominate the convergence optimization, while bi-molecular reactions (inter-molecular ineffective collision/synthesis reaction) drive the diversity search [25]. CRO has been successfully applied to various complex optimization problems, demonstrating its strong potential. For instance, Islam et al. [26] proposed a method to solve the problem of maximum coverage position by redesigning four basic CRO operators. They also introduced a repair operator to prevent the algorithm from converging to a local optimum, which increases the chance of finding the global optimum. Shaheen et al. [27] designed a hybrid method GCRO that combines CRO with greedy strategies to solve the traveling salesman problem (TSP). This method shows excellent performance in terms of error rate and execution time. Bechikh et al. [24] proposed a non-dominant CRO (NCRO), which combines the NSGA-II [5] framework with CRO. The experimental results confirm the validity of the collision operator. De et al. [28] adopted the mixed integer linear programming model to address the complex challenges of port operations and introduced the CRO algorithm to solve this problem. The experimental results show that the CRO algorithm effectively reduces the total cost and operation time. TalatHari et al. [29] developed a materials generation algorithm (MGA), which was inspired by the structure and chemical reactions of compounds to generate new materials. Its application in the optimal design of engineering problems proves the competitive advantage of MGA.

Though CRO is effective in multi-objective optimization, it has limitations in constrained scenarios. Existing CRO variants lack an effective constraint processing mechanism. They struggle to balance objective optimization and feasibility convergence, which causes search conflicts due to poor reaction collaboration. A single population structure cannot adapt well to high-dimensional constraint spaces. Thus, this paper proposes innovations based on CRO’s core reaction mechanism that introduces a dual-population and dual-stage framework. The uni/bi-molecule reactions are abstracted as convergence and diversity operators, respectively. The main population optimizes objectives and constraints in stages, and the auxiliary population offers cross-domain guidance. Combined with a weak complementarity mechanism, reaction conflicts are dynamically resolved. This effectively solves traditional CRO’s bottlenecks in constraint processing and high-dimensional search.

2.2. Multi-Population Co-Evolution Constrained Optimization Method

As an effective paradigm for solving constrained multi-objective optimization problems, the multi-population cooperative method has received extensive attention in recent years. The dual-archive algorithm C-TAEA (constrained two-archive evolutionary algorithm) proposed by Li et al. drives the Pareto frontier approximation through the convergent archive (CA) and explores the unknown region through the diverse archive (DA) [30]. However, the limited generation position of the offspring leads to insufficient convergence. The two-stage framework designed by Yuan et al. learns the correlation between the constrained front CPF and the unconstrained front UPF (unconstrained Pareto front) in the early stage, and customizes strategies based on the correlation in the later stage [31]. However, its correlation assumption is often difficult to hold in actual complex problems. Although the weak collaboration mechanism between the driver archive and the ordinary archive constructed by Li et al. introduces a diversity enhancement strategy, the strict constraint processing criterion may lose potential high-quality solutions [32]. The three-population framework extended by Wang et al. increased the collaborative complexity [33], while in the two-population divide-and-conquer strategy proposed by Ming et al. the adaptive penalty function struggled to maintain diversity in the later stage of optimization [34]. The bidirectional coevolution (BiCo) algorithm proposed by Liu et al. approximates the solution set on both the feasible and infeasible regions and is particularly suitable for boundary optimization problems [35]. However, the high-frequency information exchange between populations is prone to cause search instability. Although these methods have made breakthroughs in the collaborative mechanism, they still generally face common problems such as information interaction overload, lack of stage adaptability, and insufficient search efficiency of traditional operators.

Based on multi-population coevolution analysis, this paper achieves triple innovation to overcome framework limitations. Firstly, a weak complementary collaboration mechanism is established. The main population tackles constrained problems and the auxiliary population handles unconstrained ones, with one-way information transmission through offspring. This retains infeasible solutions’ guiding values and reduces infeasible domain traps. Secondly, a dynamic dual-stage strategy is built. In the global exploration stage, the main population focuses on objective optimization, while the auxiliary populations provide diversity support. It switches to the local development stage upon detecting stagnation, with the main population prioritizing constraints and the auxiliary population aiding fine search. Stage transitions are triggered by the population state for precise collaboration. Finally, by integrating chemical reaction optimization, a diversity-focused bi-molecular reaction aids global exploration, and a convergence-focused uni-molecular reaction enhances local development. Reaction type isolation and energy management avoid search conflicts. Compared to existing frameworks, this method improves information interaction and stage adaptability.

3. The Proposed Algorithm

3.1. Improvement of Molecular Collisions

CRO achieves a dynamic balance between global exploration and local development through four basic chemical collision reactions to enhance and optimize performance. However, during the actual execution of the algorithm various chemical reactions are prone to effect conflicts due to inconsistent action directions, which directly leads to the destruction of the historical optimal solution already obtained in the algorithm. This not only leads to a waste of computing power for the algorithm to search in the local area but also has an adverse impact on the convergence process of the overall population. To this end, this paper introduces a structure repair strategy to make full use of the local optimal solution information. And the directional search operators are designed to avoid the search conflict caused by the continuous execution of each reaction.

3.1.1. Introduction of Structural Repair Strategy

When the CRO algorithm performs decomposition or synthesis reactions, due to the immediate replacement and selection of parent and offspring individuals it fails to effectively retain or guide the historical optimal structure. Therefore, a structure repair strategy can be introduced after the decomposition and synthesis reactions to establish the connection between the historical optimal structures of the offspring and the parent generation, thereby improving the utilization efficiency of the existing solutions by the algorithm and enhancing the convergence and search stability of the population.

When the decomposition reaction occurs, the parent $S^t$ generates two random offspring individuals, $S_1^{t+1}$ and $S_2^{t+1}$, as shown in the following equation:

$$\begin{array}{c}\hfill {\mathit{S}}^{\mathit{t}}\to S_1^{t+1}+S_2^{t+1}\end{array}$$

(1)

when introducing the repair strategy, the optimal structure $S_{best}^t$ found by the parent $S^t$ is used to repair the child structure, as shown in the following formula:

$$\left\{\begin{array}{c}{S}_1^{t+1}=S_1^{t+1}+F\times S_{best}^t\hfill \\ S_2^{t+1}=S_2^{t+1}+F\times S_{best}^t\hfill \end{array}\right.$$

(2)

When the synthesis reaction occurs, two parent molecules, $S_1^t$ and $S_2^t$, generate a random offspring individual $S^{t+1}$, as shown in the following equation:

$$S_1^t+S_2^t\to S^{t+1}$$

(3)

When introducing the repair strategy, the optimal structures of the parent generation, $S_{1\_best}^t$ and $S_{2\_best}^t$, are simultaneously used to repair the structures of the child generation. The repair of the child generation $S^{t+1}$ is shown as follows:

$$S^{t+1}\to S^{t+1}+F\times (S_{1\_best}^t-S_{2\_best}^t)$$

(4)

The above F is the scaling factor ${\displaystyle \frac{t}{T_{max}}}$, t is the population algebra, and $T_{max}$ is the maximum algebra. This means that during the evolutionary process, F can adaptively adjust the degree of dependence of the offspring on the optimal structure of the parent. The closer to the late evolutionary period, the more valuable the paternal optimal structure. Therefore, the introduction of the structure repair strategy enables the search results in the evolutionary process to be fully utilized.

3.1.2. Design of Directional Search Operators

The four basic collision reactions of CRO can be classified into two categories according to their functions: promoting local domain search or global diverse search. The on-wall ineffective collisions and decomposition reactions are uni-molecular reactions, and the inter-molecular ineffective collisions and synthesis reactions are bi-molecular reactions. This means that the independent use of uni-molecular and bi-molecular reactions can also ensure the basic search capabilities required for population evolution. In addition, isolating the uni-molecular and bi-molecular reactions can prevent the algorithm from relying on the $MoleColl$ parameter and improve the applicability of the algorithm. When CRO is executed, it determines the type of reaction by generating a random number b. A uni-molecular reaction occurs when $b>MoleColl$; otherwise, a bi-molecular reaction occurs [24]. The frequency of each collision reaction is greatly influenced by the $MoleColl$ parameter, which is often difficult to determine, and a fixed parameter value may not fit the search requirements.

Therefore, two directional search operators are designed in this paper, namely, the convergence collision operator based on uni-molecule reactions and the diversity collision operator based on bi-molecule reactions. Among them, the convergence collision operator is mainly based on on-wall ineffective collision reactions and supplemented by decomposition reactions, while the diversity collision operator is mainly based on synthesis reactions and supplemented by inter-molecular ineffective collision reactions. The design abandons the original $MoleColl$ parameter setting and adopts a uniform response type execution within the population. By isolating uni-molecular and bi-molecular reactions, the problem of effect conflicts between reactions can be effectively alleviated, and the search efficiency of each reaction itself can be improved. And at the same time, the applicability and stability of the algorithm in different optimization problems can be enhanced.

3.2. Procedure of the Proposed DDCRO

Figure 1 shows the algorithm framework of DDCRO. The core lies in its integration of the CRO chemical collision reaction and the two-stage dual-population mechanism. The two cooperative populations adopt different offspring generation operators and selection strategies in different search stages, which enhances the adaptability and optimization ability of the algorithm in different search stages. The entire search process of the algorithm is divided into two different stages, and different offspring generation strategies and environment selection strategies are adopted in each stage. Specifically, in the first stage, the population focuses on global search to ensure that multiple feasible regions are covered as much as possible. Therefore, the diversity collision operator is used to help the population generate diverse solutions. Meanwhile, in this stage, the main population pays more attention to the optimization of the objective in the environmental selection strategy and relaxes the constraints to generate infeasible solutions. This promotes the global search of the population and avoids falling into a local optimum at the same time. When the population tends to be stable, it is distributed near each feasible region. At this point, it turns to the second stage. The population focuses on conducting a fine search near the feasible region to approximate the CPF. Therefore, the convergence collision operator is used to help the population optimize convergence. Meanwhile, at this stage, the main population pays more attention to constraint satisfaction in the environmental selection strategy, which ensures that the solutions in the population are all in the feasible region. Furthermore, in order to increase the diversity of the population and explore the feasible regions more effectively, DDCRO adopts a dual-population collaborative mechanism. The main population is used to solve the original constrained optimization problem and is iterated based on the feasibility constraint. While the auxiliary population is used to solve the corresponding unconstrained optimization problem, its updates are not subject to the feasibility constraint of the solution. The two populations share information to further enhance the adaptability of the populations in the search process.

The specific process of DDCRO is shown in Algorithm 1. First, DDCRO initializes the main population $Pop1$ and the auxiliary population $Pop2$ as N individuals. After the initialization is completed, the populations enter the first stage. The two populations use the diversity collision operator to generate the offspring populations $Q1$ and $Q2$, with the aim of searching the solution space as comprehensively as possible, which improves the distribution of the populations and prevents them from getting stuck in the local optimal region too early. Subsequently, the generated offspring merge with their respective parent populations, and information exchange between populations is achieved through a weak complementarity mechanism. It should be noted that the weak complementarity mechanism at this stage only affects the sharing of offspring individuals and does not involve direct substitution of the parent population. When the population evolution in the first stage tends to be stable, one must switch to the second stage for optimization. At this stage, both populations adopt the convergence collision operator for offspring generation. The key point is to guide the population to conduct a local fine search near the feasible area and gradually approach the CPF. The individual interaction strategy in the second stage remains consistent with that in the first stage. A detailed description of the stage transformation strategy is given in Section 3.3.

During the environmental selection phase, the offspring individuals generated by $Pop1$ and $Pop2$ will be screened using different environmental selection strategies, and the dominant solution will be selected based on different criteria. Specifically, $Pop1$ selects individuals based on the diversity, convergence, and feasibility of the population, but focuses on objective optimization in the first stage and constraint satisfaction in the second stage. Meanwhile, $Pop2$ was screened based on Pareto dominance and population density in these two stages. Detailed descriptions of these are provided in Section 3.4. Finally, the algorithm outputs the main population $Pop1$.

Algorithm 1 Main Procedure of DDCRO

Input:  N (size of populations), $T_{max}$ (maximum generation), $l_{gap}$ (generation gap), $\delta$ (a positive threshold) Output:  the final $Pop1$   1: $Pop1$, $Pop2$ ← Initialization (N);   2: $Stage$ ← $\mathrm{ }true$;   3: while$t\le T_{max}$do   4:    // In Stage 1   5:    if $Stage==true$ then   6:        $MatingPool1$ ← Select N parents from $Pop1$;   7:        $MatingPool2$ ← Select N parents from $Pop2$;   8:        $Q1$ ← Generate $N/2$ offspring based on $MatingPool1$ by uni-molecular operators;   9:        $Q2$ ← Generate $N/2$ offspring based on $MatingPool2$ by uni-molecular operators; 10:        $Pop1$ ← $Pop1\cup Q1\cup Q2$; 11:        $Pop2$ ← $Pop2\cup Q1\cup Q2$; 12:        $Pop1$ ← Compare the solutions in $Pop1$ and select N solutions by stage1-Pop1 strategy; 13:        $Pop2$ ← Compare the solutions in $Pop2$ and select N solutions by Pop2 strategy; 14:        if $t>l_{gap}$ then 15:            Calculate the change rate r in $Pop1$ according to Equations (5) and (6); 16:            if $r\le \delta$ then 17:                $Stage$ ← $false$; // switch to stage 2 18:            end if 19:         end if 20:      end if 21:      // In Stage 2 22:      if $Stage==false$ then 23:          $MatingPool1$ ← Select $2N$ parents from $Pop1$; 24:          $MatingPool2$ ← Select $2N$ parents from $Pop2$; 25:          $Q1$ ← Generate N offspring based on $MatingPool1$ by bi-molecular operators; 26:          $Q2$ ← Generate N offspring based on $MatingPool2$ by bi-molecular operators; 27:          $Pop1$ ← $Pop1\cup Q1\cup Q2$; 28:          $Pop2$ ← $Pop2\cup Q1\cup Q2$; 29:          $Pop1$ ← Compare the solutions in $Pop1$ and select N solutions by stage2-Pop1 strategy; 30:          $Pop2$ ← Compare the solutions in $Pop2$ and select N solutions by Pop2 strategy. 31:      end if 32: end while

3.3. The Mechanism of Two-Stage

DDCRO adopts a phased optimization strategy to divide the evolutionary process into two stages. The main goal of the first stage is that the population can search for as many feasible regions as possible when facing a broad feasible region. When the population converges to the vicinity of the feasible region and it is difficult for the population to undergo significant changes, the population shifts to the second stage. At this stage, the population needs to develop the neighborhood more precisely near the feasible region to further approach the CPF. Different collision operators and environmental selection strategies are adopted at different stages, which aim to enhance the adaptability of the population during the search process. Specifically, in the diversity collision operators used in the first stage, the population is controlled to optimize towards global diversity based on the combination of BLX-$\alpha$ and GM operators and synthesis conditions. At this stage, the population focuses on objective optimization. In the convergence collision operators used in the second stage, the local convergence of population is controlled by combining PM and GM operators with decomposition conditions. At this stage, the population focuses on constraint satisfaction.

When the main population $Pop1$ stalls, it is necessary to switch to the second stage. In DDCRO, the changes of ideal point, average point, and nadir point are mainly used to measure the conditions of phase transition [36]. They are respectively expressed as $z_k^i=(z_1^t,\dots ,z_m^t)$, $a_k^i=(a_1^t,\dots ,a_m^t)$ and $n_k^i=(n_1^t,\dots ,n_m^t)$, where i is the current generation and m is the number of objectives. The rate of change of the three points from the ($i-l_{gap}$) generation to the i generation can be expressed as:

$$\left\{\begin{array}{c}{r}_z^i=\underset{k=1,\dots ,m}{max}\{{\displaystyle \frac{|z_k^i-z_k^{i-l_{gap}}|}{max\{|z_k^{i-l_{gap}}|,\mathrm{\Delta }\}}}\}\hfill \\ r_a^i=\underset{k=1,\dots ,m}{max}\{{\displaystyle \frac{|a_k^i-a_k^{i-l_{gap}}|}{max\{|a_k^{i-l_{gap}}|,\mathrm{\Delta }\}}}\}\hfill \\ r_n^i=\underset{k=1,\dots ,m}{max}\{{\displaystyle \frac{|n_k^i-n_k^{i-l_{gap}}|}{max\{|n_k^{i-l_{gap}}|,\mathrm{\Delta }\}}}\}\hfill \end{array}\right.$$

(5)

where $\mathrm{\Delta }$ is a small positive number to prevent the denominator from being zero and $l_{gap}$ represents the interval number of generations. This is used to ensure that the population undergoes a sufficient number of iterations in the first stage to prevent the population from stagnating too quickly. The setting of $l_{gap}$ only needs to be kept within a reasonable range, and its value is set to 20 in this paper. After obtaining the rates of change of the three points, $r^t$ is calculated as follows:

$$r^t=max\{r_z^i,r_a^i,r_n^i\}$$

(6)

and $r^t$ is then compared with the threshold $\delta$ to determine the current search stage.

The threshold $\delta$ is set to ${\displaystyle \frac{t\times P_{feasible}}{T_{max}}}$, and $P_{feasible}$ represents the proportion of the feasible region in the entire solution space. This adaptive adjustment method of $\delta$ can ensure the reliability of the timing of stage transitions. First of all, algorithms are more likely to enter the second stage in the late evolutionary period. In addition, the threshold also matches the complexity of the feasible domain of the problem. For complex constraint problems with a small proportion of feasible regions, the stage transition can be delayed in order to fully explore. The algorithm will continue to execute the first stage when $r^t>\delta$, in which the diversity collision operator is used to increase the probability of finding the feasible region and continuously approach the CPF. When $r^t\le \delta$, it can be inferred that the search process has reached a stable state and the population is near the CPF. At this time, the exploration of the current area begins to approximate the entire CPF.

3.4. The Mechanism of Dual Population

DDCRO employs the dual-population mechanism at each stage, aiming to fully utilize the advantages of feasible and infeasible solutions in the evolution process. When dealing with CMOPs, the feasible solution regions are often separated by infeasible regions. The information from infeasible solutions can greatly help the population escape the local optimum. Therefore, an auxiliary population needs to be designed to search for infeasible solutions and the main population is used to solve the CMOP. A weak complementarity mechanism is adopted among the populations, and the feasible solutions in the main population can help the auxiliary population evolve towards the feasible region. Meanwhile, the infeasible solutions found by the auxiliary population can promote the individuals of the main population to jump out of the local optimum in time. When dealing with constraint problems, this weak complementarity mechanism has been proven to be a more effective way compared to strong complementarity [37].

The two populations handle constrained and unconstrained problems, respectively, which also determines that different environmental selection strategies need to be employed. Meanwhile, the stage of the population search also needs to be considered for strategy design. Specifically, the auxiliary population $Pop2$ has the same selection strategy in both stages. The main population needs to adopt different environmental selection strategies at different search stages. In the first stage, $Pop1$ conducts a global search to allow the population to have more infeasible solutions. In the second stage, $Pop1$ conducts a local search near the CPF to emphasize the feasibility of the solution. Therefore, this paper designs three environmental selection strategies, which are respectively applied to different stages and different populations. A detailed description of these follows.

3.4.1. Stage1—Pop 1 Environmental Selection

This strategy is applied to the main population in the first stage. First of all, the convergence, diversity, and feasibility of the main population need to be considered comprehensively in environmental selection. Secondly, we need to focus on these three metrics in different stages. And in the first stage, we should ease the restrictions on their feasibility in order to promote the diversity of solutions. Therefore, it is necessary to redesign the potential energy equation to meet the characteristics of the CRO algorithm’s evolution towards low potential energy [38].

For the measure of individual convergence, it is expressed by the sum of normalized values of its objectives, as shown in the following:

$$\left\{\begin{array}{c}{\overline{f}}_j\left(p\right)={\displaystyle \frac{f_j\left(p\right)-f_{min,j}}{f_{max,j}-f_{min,j}}}\hfill \\ f_{pr}\left(p\right)=\sum _{j=1}^M{\overline{f}}_j\left(p\right)\hfill \end{array}\right.$$

(7)

where $f_{max,j}$ and $f_{min,j}$ respectively represent the maximum and minimum values of the j-th objective, and ${\overline{f}}_j\left(p\right)$ represents the normalized values. The smaller ${\overline{f}}_j\left(p\right)$ is, the better the convergence of the solutions is.

Individual diversity is measured by evaluating both inter-solution similarity and spatial distribution within the objective space, as defined in Equation (8):

$$f_{cd}\left(p\right)=\sqrt{\sum _{q\in X,q\ne p}sh(p,q)}$$

(8)

where $sh(p,q)$ represents the shared function between individuals p and q and is defined as follows:

$$sh=\left\{\begin{array}{cc}{(0.5\times (1-{\displaystyle \frac{d(p,q)}{r}}))}^2,\hfill & ifd(p,q)<r, f_{pr}\left(p\right)<f_{pr}\left(q\right)\hfill \\ {(1.5\times (1-{\displaystyle \frac{d(p,q)}{r}}))}^2,\hfill & ifd(p,q)<r, f_{pr}\left(p\right)>f_{pr}\left(q\right)\hfill \\ rand\left(\right),\hfill & ifd(p,q)<r, f_{pr}\left(p\right)=f_{pr}\left(q\right)\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.$$

(9)

where $d(p,q)$ represents the Euclidean distance of individuals p and q in the objective space, $rand\left(\right)$ represents randomly selecting one of the shared functions for calculation, and r represents the regional radius, which is determined by the population size N and the objective number M. The calculation formula of r is as follows:

$$r={\displaystyle \frac{1}{\sqrt[M]{N}}}$$

(10)

$f_{cd}\left(p\right)$ is smaller when the Euclidean distance between individuals is larger, which means that individual diversity is better.

Based on the above considerations, we employ $min\{f_{pr}\left(p\right),f_{cd}\left(p\right)\}$ for Pareto non-dominated sorting to determine the ranking $rank\left(p\right)$ of each individual, and we formulate the final $PE$ equation as follows:

$$PE\left(p\right)=rank\left(p\right)+{\displaystyle \frac{f_{cv}\left(p\right)}{f_{cv}\left(p\right)+1}}$$

(11)

where $rank\left(p\right)$ is an integer, ensuring that the formula is still dominated by the $rank\left(p\right)$ value. This aligns with the current stage’s goal of having the main population focus on optimizing objectives with relaxed constraints. When the rankings are consistent, the smaller the $f_{cv}$, the smaller the $PE$, which meets the requirements of the preference for superior individuals with lower $PE$ values.

The main process of the environment selection strategy is shown in Algorithm 2. The parent $Pop1$, the offspring $Q1$ of the main population, and the offspring $Q2$ of the auxiliary population are combined through the weak complementarity mechanism to obtain a combined population $CP$. The convergence, diversity, and feasibility of each individual in this population are calculated respectively, and non-dominated sorting is applied to determine ranking values. Finally, the $PE$ of each individual is calculated by the constructed potential energy equation, and the N individual with the smallest $PE$ is selected to enter the next iteration.

Algorithm 2 Stage1-Pop1 Environmental Selection

Input:  $Pop1$ (the main population parents), $Q1$ (the main population offspring), $Q2$ (the auxiliary population offspring) Output:  $Pop1$   1: $CP$ ← $Pop1\cup Q1\cup Q2$;   2: for each individual $p\in CP$ do   3:     $f_{pr}\left(p\right)$ ← calConvergence (p);   4:     $f_{cd}\left(p\right)$ ← calDiversity (p);   5:     $f_{cv}\left(p\right)$ ← calFeasibility (p);   6: end for   7: $rank\left(p\right)$ ← ParetoNodominatedSort ($f_{pr}\left(p\right)$,$f_{cd}\left(p\right)$);   8: $PE\left(p\right)$ ← $rank\left(p\right)$ + ${\displaystyle \frac{f_{cv}\left(p\right)}{f_{cv}\left(p\right)+1}}$;   9: $Pop1$ ← Select N individuals with smaller $PE$ from $CP$.

3.4.2. Stage2—Pop1 Environmental Selection

This strategy is implemented for the main population in the second stage. During this phase, all solutions must remain within the feasible region to guarantee that the output population satisfies all constraint conditions. Consequently, the selection strategy prioritizes constraint satisfaction. For measuring individual convergence and diversity, we employ the PE definition from NCRO, as described below:

$$PE\left(p\right)=rank\left(p\right)+e^{-crowd\left(p\right)}$$

(12)

The individual feasibility is still measured by $f_{cv}$, but a certain priority control strategy is needed when selecting individuals, as shown in Algorithm 3. The parent $Pop1$, the offspring $Q1$ of the main population, and the offspring $Q2$ of the auxiliary population are combined through the weak complementarity mechanism to obtain a combined population, $CP$. Then, the constraint violation degree $f_{cv}\left(p\right)$ of each individual is calculated. According to this value, the population is divided into feasible solution population $FS$ and infeasible solution population $IFS$, and the solutions in the two populations are sorted in ascending order by $PE$ value. If the number of individuals in $FS$ exceeds the population size N, the top N individuals are selected to enter the next generation. On the contrary, all individuals in $FS$ are combined with individuals in $IFS$ with smaller $PE$ to form a population with size N and enter the next generation.

Algorithm 3 Stage2-Pop1 Environmental Selection

Input:  $Pop1$ (the main population parents), $Q1$ (the main population offspring), $Q2$ (the auxiliary population offspring), M (number of objectives), N (population size) Output:  $Pop1$   1: $FS$,$IFS$ ←∅;   2: $CP$ ← $Pop1\cup Q1\cup Q2$;   3: Calculate $PE$ of each individual of $CP$;   4: for each individual $p\in CP$ do   5:     $f_{cv}\left(p\right)$ ← calFeasibility (p);   6:     if $f_{cv}\left(p\right)$ = 0 then   7:        Add p into $FS$; // add to feasible solution set   8:     else   9:        Add p into $IFS$; // add to infeasible solution set 10:     end if 11: end for 12: $newPop1$ ←∅; 13: Sort $FS$,$IFS$ by $PE$ in ascending order; 14: if $len\left(FS\right)$ ≥ N then 15:     $newPop1$ ← Select the first N individuals from $FS$; 16: else 17:     $newPop1$ ← Select all individuals from $FS$; 18: end if 19: $newPop1$ ← $newPop1\cup IFS[0:N-len(FS\left)\right]$; 20: $Pop1$ ← $newPop1$.

3.4.3. Pop2 Environmental Selection

This strategy is applied to the auxiliary population in the first and second stages. The population optimizes the unconstrained problem to share the infeasible solution with the main population, so as to promote the rapid convergence and uniform distribution of the main population. It is necessary to help the population converge to the unconstrained PF in the environmental selection strategy. Therefore, we adopt the original NCRO’s definition of $PE$, which prioritizes individual ranking before considering population crowding degree. This idea plays a significant role in solving low-dimensional MOPs, and its calculation is shown in Equation (12). The individuals with smaller potential energy will be selected to enter the next generation.

3.5. Computational Complexity

Assume that both populations have a size of N, an objective number of M, and a constraint number of q. The algorithm complexity of DDCRO is considered in two stages. In the first stage, the complexity of parent selection and offspring generation of the two populations is $O\left(N\right)$. The main population adopts the three-index evaluation method, and its computational complexity is $O(M{N}^2+qN)$. The Pareto-based non-dominated sorting complexity is $O\left(M{N}^2\right)$. The environmental selection strategy of the auxiliary population adopts the potential energy equation in the NCRO algorithm, and its complexity is $O\left(MNlogN\right)$. So the total time complexity of the first stage is $O(M{N}^2+qN)$. In the second stage, the complexity of parent selection and offspring generation of the two populations is $O\left(N\right)$. In terms of environment selection strategy, the main population computes the PE using NCRO’s potential energy equation and incorporates the calculation of individual constraint violation degrees, resulting in a time complexity of $O(MNlogN+qN)$. The complexity of the auxiliary population is the same as that in the first stage, and the total time complexity in the second stage is $O(MNlogN+qN)$. Therefore, the total time complexity of DDCRO is $O(M{N}^2+qN)$. The theoretical complexity aligns with established excellent multi-objective optimization algorithms like NSGA-II [5]. Notably, while maintaining equivalent computational complexity, subsequent experiments verify its superiority in complex constraint scenarios. Furthermore, the linear scaling of constraint processing cost ($O\left(qN\right)$) suggests that the algorithm can efficiently accommodate additional constraints without incurring exponential overhead. In summary, the proposed DDCRO shows significant advantages in terms of computational efficiency and scalability.

4. Experimental Studies

4.1. Experimental Settings

4.1.1. Test Problems

A total of four benchmark test suites were selected for the experiments—DTLZ [39], MW [40], DOC [41], and LIRCMOP [42]—based on their unique constraint features and optimization challenges. The DTLZ test suite includes problems like C2-DTLZ2 and DC1-DTLZ, which feature both continuous and discontinuous CPFs, along with high-dimensional infeasible regions exemplified by DC2-DTLZ1/3. These characteristics demand algorithms to balance exploration and exploitation within fragmented feasible regions, while simultaneously avoiding convergence to local optima. The MW test suite is characterized by the diversity of its feasible regions. Specifically, it includes narrow and discrete regions in MW1/5/6, discontinuous regions in MW7/10/11, and large-area discontinuous regions in MW8/14. These challenges require algorithms to maintain solution diversity in sparsely distributed feasible spaces and adapt to multi-modal constraint landscapes. The DOC test suite imposes dual constraints in decision and objective spaces, creating extremely narrow feasible regions like DOC3/6 with mixed CPF characteristics. It can test the convergence of the algorithm under strict constraints. The LIRCMOP test suite focuses on high-dimensional problems and spatial dislocation between UPF and CPF, which includes discontinuous line-segment CPFs (LIRCMOP1-4) and large-area discrete CPFs (LIRCMOP9-12). These problems demand robust CPF approximation in fragmented spaces with extreme infeasible regions. The parameter settings for these test problems are presented in

Table 1. Parameter settings of test suites.

Test Suite	Test Problem	Objectives	Variables
DTLZ	C1/DC1/DC2/DC3-DTLZ1	3	7
	others	3	12
MW	MW1–14	2	15
DOC	DOC1	2	6
	DOC2	2	16
	DOC3	2	10
	DOC4–5	2	8
	DOC6–7	2	11
	DOC8	2	10
	DOC9	2	11
LIRCMOP	LIRCMOP1–12	2	10
	LIRCMOP13–14	3	10

.

Additionally, to substantiate the applicability of the DDCRO algorithm in real-world scientific and engineering scenarios, we have meticulously selected eight distinct CMOPs from practical applications for testing the real scene. These encompass the disc brake design problem, gear train design problem, four-bar plane truss, two-bar plane truss, water resource management problem, grash energy management for high-speed train, process synthesis problem, and process flow sheeting problem. All these problems are sourced from the RWMOP test suite, which spans a wide array of domains. And they incorporate a diverse range of real-world scenarios, with detailed definitions provided in reference [43].

4.1.2. Comparison Algorithms

To evaluate the performance of the DDCRO algorithm, six constrained multi-objective optimization algorithms were selected as comparison benchmarks, namely, NSGA-II [5], CAEAD [44], CCMO [37], URCMO [8], TSTI [17], and CMODE-FTR [45]. NSGA-II employs the CDP strategy to implement a feasible solution prioritization mechanism. TSTI designs a phased optimization strategy to dynamically adjust environmental selection criteria at different stages. CMODE-FTR innovatively integrates CDP with Pareto non-dominated sorting to construct a dynamic evaluation index. The other three comparison algorithms all adopt a dual-population mechanism. Specifically, CAEAD implements an evolution/degradation strategy based on the distribution characteristics of the auxiliary population, which enhances the information transfer efficiency to the main population through the state of the auxiliary population. CCMO establishes a weak complementary offspring sharing mechanism between the main population (leading constrained optimization) and the auxiliary population (focusing on auxiliary problem-solving). URCMO dynamically classifies problem types based on the spatial relationship between the PF and the CPF, and accordingly customizes the evolutionary strategy for the auxiliary population.

To ensure fairness, a unified parameter configuration was adopted in the experiments. The distribution index and probability of the SBX operator were set to 20 and ${\displaystyle \frac{1}{n}}$. The mutation index and probability of the polynomial mutation operator were set to 20 and 0.9. The $\alpha$ parameter in the BLX-$\alpha$ crossover operator was fixed at 0.5. The crossover rate ($CR$) and scaling factor (F) parameters in the differential evolution operator were set to 1 and 0.5. The population size (N) was set to 100, and the maximum number of function evaluations ($maxFE$) was set to 100,000. All algorithms were independently run 30 times to eliminate the influence of randomness, and the remaining parameters of the comparison algorithms were kept consistent with those in the original paper.

4.1.3. Assessment Criteria

This paper employs two comprehensive indicators to evaluate the comprehensive performance of the algorithm: the inverted generational distance (IGD) metric and the hypervolume (HV) evaluation metric. The IGD metric quantifies the convergence of the solution set obtained by the algorithm in approximating the true PF. It is defined as:

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

(13)

where P represents a set of reference points uniformly sampled from the true PF, $\left|P\right|$ denotes the sampling size, and Q is the optimal solution set evolved by the algorithm. $d(v,Q)$ is the minimum Euclidean distance from the reference point v in P to the population Q. A smaller IGD value indicates better individual performance. The HV metric measures the dominated volume of the solution set Q in the objective space. It is calculated as follows:

$$HV=vol ({\cup }_{i\in Q}[f_1\left(i\right),r_1]\times \dots \times [f_m\left(i\right),r_m])$$

(14)

where $R=(r_1,r_2,\dots ,r_m)$ is a set of reference points in the objective space and Q is the optimal solution set output by the algorithm. A larger HV value indicates stronger convergence and diversity of the solution set.

In addition, the experiment are conducted on the open-source evolutionary multi-objective optimization platform named PlatEMO [46], and utilizes the Wilcoxon rank-sum test with a significance level of 0.05 to observe whether or not there are significant differences between the experimental results of the two algorithms. The symbols “+”, “− ”, and “=” are used to denote that the performance of the comparison algorithm is significantly better than, worse than, or equivalent to the performance of the algorithm proposed in this paper, respectively. And “NaN” indicates that no solution of the comparison algorithm converges near CPF.

4.2. Experimental Results and Analysis

4.2.1. Comparisons on DTLZ

To evaluate the comprehensive performance of the DDCRO algorithm in complex constraint scenarios, this experiment selected the DTLZ test suite, which encompasses a variety of constraint characteristics. The focus was on examining the algorithm’s adaptability to key challenges such as obstacles in infeasible regions, discontinuous CPF, and high-dimensional infeasible regions.

Table 2. IGD values and HV values obtained by seven algorithms on DTLZ test suite.

Problem		NSGAII	CAEAD	CCMO	URCMO	TSTI	CMODE-FTR	DDCRO
C1_DTLZ1	IGD	2.7961$\times {10}^{-2}$ (4.50$\times {10}^{-4}$)−	2.2881$\times {10}^{-2}$ (1.43$\times {10}^{-4}$)−	1.9909$\times {10}^{-2}$ (1.33$\times {10}^{-4}$)≈	2.2185$\times {10}^{-2}$ (6.81$\times {10}^{-5}$)−	2.0018$\times {10}^{-2}$ (7.80$\times {10}^{-5}$)−	2.6349$\times {10}^{-2}$ (0.00$\times {10}^0$)≈	1.9655$\times {10}^{-2}$ (1.19$\times {10}^{-4}$)
	HV	8.1577$\times {10}^{-1}$ (1.12$\times {10}^{-2}$)−	8.3624$\times {10}^{-1}$ (6.34$\times {10}^{-4}$)−	8.3762$\times {10}^{-1}$ (1.40$\times {10}^{-3}$)≈	8.2152$\times {10}^{-1}$ (4.10$\times {10}^{-3}$)−	8.4064$\times {10}^{-1}$ (1.91$\times {10}^{-3}$)≈	8.2191$\times {10}^{-1}$ (0.00$\times {10}^0$)≈	8.3811$\times {10}^{-1}$ (9.97$\times {10}^{-4}$)
C1_DTLZ3	IGD	6.0485$\times {10}^0$ (3.96$\times {10}^0$)−	2.8450$\times {10}^0$ (3.65$\times {10}^0$)−	5.4115$\times {10}^{-2}$ (5.31$\times {10}^{-4}$)≈	1.5810$\times {10}^{-1}$ (1.13$\times {10}^{-1}$)−	8.0147$\times {10}^0$ (5.65$\times {10}^{-3}$)−	2.3152$\times {10}^0$ (3.85$\times {10}^0$)−	5.3272$\times {10}^{-2}$ (8.45$\times {10}^{-4}$)
	HV	1.1382$\times {10}^{-1}$ (2.28$\times {10}^{-1}$)−	7.2555$\times {10}^{-2}$ (1.27$\times {10}^{-1}$)−	5.5600$\times {10}^{-1}$ (8.40$\times {10}^{-4}$)≈	3.7335$\times {10}^{-1}$ (1.91$\times {10}^{-1}$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	7.7162$\times {10}^{-2}$ (1.08$\times {10}^{-1}$)−	5.5878$\times {10}^{-1}$ (3.02$\times {10}^{-3}$)
C2_DTLZ2	IGD	5.6409$\times {10}^{-2}$ (1.98$\times {10}^{-3}$)−	5.5431$\times {10}^{-2}$ (2.69$\times {10}^{-3}$)−	4.2781$\times {10}^{-2}$ (9.73$\times {10}^{-4}$)≈	4.3246$\times {10}^{-2}$ (9.06$\times {10}^{-4}$)≈	4.2097$\times {10}^{-2}$ (5.55$\times {10}^{-4}$)≈	5.5665$\times {10}^{-2}$ (1.62$\times {10}^{-3}$)−	4.2093$\times {10}^{-2}$ (3.17$\times {10}^{-4}$)
	HV	4.8434$\times {10}^{-1}$ (2.95$\times {10}^{-3}$)−	5.0360$\times {10}^{-1}$ (4.35$\times {10}^{-3}$)−	5.1557$\times {10}^{-1}$ (8.27$\times {10}^{-4}$)−	5.1180$\times {10}^{-1}$ (2.93$\times {10}^{-3}$)−	5.1571$\times {10}^{-1}$ (2.38$\times {10}^{-3}$)≈	4.8158$\times {10}^{-1}$ (5.67$\times {10}^{-3}$)−	5.1864$\times {10}^{-1}$ (9.17$\times {10}^{-4}$)
C3_DTLZ4	IGD	1.2544$\times {10}^{-1}$ (4.08$\times {10}^{-3}$)−	1.1157$\times {10}^{-1}$ (2.17$\times {10}^{-3}$)−	9.5734$\times {10}^{-2}$ (1.34$\times {10}^{-3}$)≈	1.0654$\times {10}^{-1}$ (2.34$\times {10}^{-3}$)−	9.6538$\times {10}^{-2}$ (9.78$\times {10}^{-4}$)≈	1.3089$\times {10}^{-1}$ (3.73$\times {10}^{-3}$)−	9.5405$\times {10}^{-2}$ (8.10$\times {10}^{-4}$)
	HV	7.6724$\times {10}^{-1}$ (4.94$\times {10}^{-3}$)−	7.8501$\times {10}^{-1}$ (1.03$\times {10}^{-3}$)−	7.8922$\times {10}^{-1}$ (6.32$\times {10}^{-4}$)≈	7.8134$\times {10}^{-1}$ (2.23$\times {10}^{-3}$)−	7.8923$\times {10}^{-1}$ (7.76$\times {10}^{-4}$)≈	7.6839$\times {10}^{-1}$ (7.16$\times {10}^{-3}$)−	7.8994$\times {10}^{-1}$ (1.39$\times {10}^{-3}$)
DC1_DTLZ1	IGD	1.4610$\times {10}^{-2}$ (6.77$\times {10}^{-4}$)−	1.5260$\times {10}^{-2}$ (2.93$\times {10}^{-4}$)−	1.1446$\times {10}^{-2}$ (7.72$\times {10}^{-5}$)≈	1.2004$\times {10}^{-2}$ (5.25$\times {10}^{-4}$)−	1.1974$\times {10}^{-2}$ (9.56$\times {10}^{-4}$)≈	1.1554$\times {10}^{-1}$ (2.01$\times {10}^{-1}$)−	1.1432$\times {10}^{-2}$ (9.27$\times {10}^{-5}$)
	HV	6.1347$\times {10}^{-1}$ (3.67$\times {10}^{-3}$)−	6.2671$\times {10}^{-1}$ (9.53$\times {10}^{-4}$)−	6.3070$\times {10}^{-1}$ (9.90$\times {10}^{-4}$)≈	6.2819$\times {10}^{-1}$ (1.69$\times {10}^{-3}$)−	6.2738$\times {10}^{-1}$ (5.86$\times {10}^{-3}$)≈	4.5649$\times {10}^{-1}$ (3.02$\times {10}^{-1}$)−	6.3171$\times {10}^{-1}$ (8.95$\times {10}^{-4}$)
DC1_DTLZ3	IGD	4.4127$\times {10}^{-2}$ (3.56$\times {10}^{-3}$)−	8.9060$\times {10}^{-1}$ (1.62$\times {10}^0$)−	3.4245$\times {10}^{-2}$ (7.21$\times {10}^{-4}$)≈	7.9188$\times {10}^{-2}$ (4.95$\times {10}^{-2}$)≈	8.1523$\times {10}^{-2}$ (5.92$\times {10}^{-2}$)≈	1.0252$\times {10}^0$ (1.09$\times {10}^0$)−	3.5488$\times {10}^{-2}$ (1.14$\times {10}^{-3}$)
	HV	4.5791$\times {10}^{-1}$ (5.09$\times {10}^{-3}$)−	2.8181$\times {10}^{-1}$ (2.18$\times {10}^{-1}$)−	4.6879$\times {10}^{-1}$ (4.06$\times {10}^{-3}$)≈	3.7338$\times {10}^{-1}$ (1.07$\times {10}^{-1}$)≈	3.8028$\times {10}^{-1}$ (1.08$\times {10}^{-1}$)≈	8.9104$\times {10}^{-2}$ (1.77$\times {10}^{-1}$)−	4.6700$\times {10}^{-1}$ (3.94$\times {10}^{-3}$)
DC2_DTLZ1	IGD	NaN (NaN)	3.7956$\times {10}^{-2}$ (2.98$\times {10}^{-2}$)−	2.0391$\times {10}^{-2}$ (4.14$\times {10}^{-4}$)≈	3.1066$\times {10}^{-2}$ (8.71$\times {10}^{-3}$)−	2.8059$\times {10}^{-2}$ (0.00$\times {10}^0$)≈	NaN (NaN)	2.0044$\times {10}^{-2}$ (4.38$\times {10}^{-4}$)
	HV	NaN (NaN)	7.9008$\times {10}^{-1}$ (9.61$\times {10}^{-2}$)≈	8.3990$\times {10}^{-1}$ (1.71$\times {10}^{-3}$)≈	8.0620$\times {10}^{-1}$ (2.54$\times {10}^{-2}$)−	8.1348$\times {10}^{-1}$ (0.00$\times {10}^0$)≈	NaN (NaN)	8.4112$\times {10}^{-1}$ (3.23$\times {10}^{-3}$)
DC2_DTLZ3	IGD	NaN (NaN)	5.6924$\times {10}^{-1}$ (9.65$\times {10}^{-4}$)≈	5.3675$\times {10}^{-2}$ (6.31$\times {10}^{-4}$)≈	5.7923$\times {10}^{-1}$ (4.28$\times {10}^{-3}$)≈	NaN (NaN)	NaN (NaN)	1.2851$\times {10}^{-1}$ (1.05$\times {10}^{-1}$)
	HV	NaN (NaN)	4.7510$\times {10}^{-3}$ (8.55$\times {10}^{-4}$)≈	4.3228$\times {10}^{-1}$ (1.61$\times {10}^{-3}$)−	1.0134$\times {10}^{-2}$ (2.21$\times {10}^{-3}$)≈	NaN (NaN)	NaN (NaN)	5.5553$\times {10}^{-1}$ (1.58$\times {10}^{-1}$)
DC3_DTLZ1	IGD	1.8108$\times {10}^{-1}$ (1.48$\times {10}^{-1}$)−	2.2275$\times {10}^{-1}$ (4.27$\times {10}^{-1}$)−	7.0380$\times {10}^{-3}$ (1.54$\times {10}^{-4}$)≈	9.7489$\times {10}^{-2}$ (1.81$\times {10}^{-1}$)≈	3.0917$\times {10}^{-1}$ (1.53$\times {10}^{-1}$)−	4.9849$\times {10}^{-1}$ (6.46$\times {10}^{-1}$)−	7.1917$\times {10}^{-3}$ (3.71$\times {10}^{-4}$)
	HV	1.8058$\times {10}^{-1}$ (2.34$\times {10}^{-1}$)−	3.9322$\times {10}^{-1}$ (2.62$\times {10}^{-1}$)≈	5.3167$\times {10}^{-1}$ (1.46$\times {10}^{-3}$)≈	3.9911$\times {10}^{-1}$ (2.66$\times {10}^{-1}$)≈	6.6111$\times {10}^{-2}$ (6.16$\times {10}^{-2}$)−	2.6532$\times {10}^{-1}$ (3.06$\times {10}^{-1}$)≈	5.3320$\times {10}^{-1}$ (4.62$\times {10}^{-3}$)
DC3_DTLZ3	IGD	1.4270$\times {10}^0$ (3.18$\times {10}^{-1}$)−	1.1605$\times {10}^0$ (2.15$\times {10}^0$)≈	2.2225$\times {10}^{-2}$ (1.58$\times {10}^{-3}$)≈	4.8606$\times {10}^{-1}$ (1.49$\times {10}^{-1}$)≈	2.7253$\times {10}^0$ (3.33$\times {10}^{-1}$)−	2.1272$\times {10}^0$ (2.69$\times {10}^0$)−	1.3855$\times {10}^{-1}$ (1.80$\times {10}^{-1}$)
	HV	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	1.9122$\times {10}^{-1}$ (1.47$\times {10}^{-1}$)≈	3.5135$\times {10}^{-1}$ (8.50$\times {10}^{-3}$)≈	2.5569$\times {10}^{-2}$ (5.11$\times {10}^{-2}$)≈	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	2.4469$\times {10}^{-1}$ (1.35$\times {10}^{-1}$)
$+/-/\approx$	0/0/20	8/12/00	2/18/00	11/9/00	9/11/00	0/17/3

* Data with a dark background shows that the algorithm has reached the optimal value for the relevant problem index.

presents the experimental results of seven algorithms on 20 test cases. From an overall performance perspective, the proposed DDCRO algorithm achieved the optimal results on 13 test problems. CCMO performed best on six test problems, CMODE-FTR excelled in one problem, and the remaining algorithms did not obtain the optimal results. Specifically, while both the DDCRO and CCMO algorithms utilize a dual-population approach, DDCRO’s two-stage optimization strategy allows it to better balance exploration and exploitation. This advantage is particularly evident in problems with significant infeasible regions, such as C1-DTLZ1 and C1-DTLZ3. In these cases, DDCRO’s auxiliary population provides valuable guidance to the main population, enabling it to traverse infeasible regions more effectively and converge towards the CPF. Comparing CMODE-FTR with DDCRO, the former dynamically adjusts the evaluation index but lacks a structured approach to handle complex constraint scenarios. DDCRO’s structured repair strategy and directional search operators ensure that high-quality solutions are preserved during reactions, leading to better performance on problems with discontinuous CPFs (e.g., C2-DTLZ2, C3-DTLZ4).

The population distribution plots are crucial for evaluating multi-objective optimization algorithms, visually presenting solution distributions while offering insights into search behaviors. Their significance lies in validating convergence and diversity, revealing the exploration and exploitation capabilities under constraints and guiding algorithmic refinements by identifying strengths and weaknesses for targeted improvements. Therefore, in order to more clearly contrast the population differences among the algorithms, Figure 2 presents the population distribution of each algorithm on the C1-DTLZ3 problem. As can be seen, due to the use of the CDP strategy, NSGA-II gets trapped in local optima and is unable to cross the infeasible region obstacles. TSTI also lacks a mechanism to utilize infeasible solution information, which leads to stagnation in population evolution. In contrast, the dual-population strategy and weak complementary mechanism employed by DDCRO and CCMO effectively help the populations cross the local optima regions and achieve good convergence to the CPF.

4.2.2. Comparisons on MW

Given the complex characteristics of the MW test suite, such as high-density infeasible-region barriers and discontinuous CPF structures, this experiment aims to verify the synergy of the algorithm in the two stages of diversity maintenance and convergence enhancement. The experimental results for each algorithm are presented in

Table 3. IGD values and HV values obtained by seven algorithms on MW test suite.

Problem		NSGAII	CAEAD	CCMO	URCMO	TSTI	CMODE-FTR	DDCRO
MW1	IGD	8.8883$\times {10}^{-2}$ (1.45$\times {10}^{-1}$)−	2.0225$\times {10}^{-3}$ (6.17$\times {10}^{-5}$)≈	1.6301$\times {10}^{-3}$ (1.29$\times {10}^{-5}$)≈	1.7771$\times {10}^{-3}$ (2.58$\times {10}^{-4}$)≈	1.6596$\times {10}^{-3}$ (5.28$\times {10}^{-5}$)≈	3.6838$\times {10}^{-3}$ (3.53$\times {10}^{-3}$)−	1.6166$\times {10}^{-3}$ (3.68$\times {10}^{-5}$)
	HV	4.1526$\times {10}^{-1}$ (1.17$\times {10}^{-1}$)≈	4.8882$\times {10}^{-1}$ (2.43$\times {10}^{-5}$)≈	4.8986$\times {10}^{-1}$ (1.02$\times {10}^{-4}$)≈	4.8971$\times {10}^{-1}$ (4.18$\times {10}^{-4}$)≈	4.8950$\times {10}^{-1}$ (6.63$\times {10}^{-4}$)≈	4.8657$\times {10}^{-1}$ (6.20$\times {10}^{-3}$)≈	4.8969$\times {10}^{-1}$ (2.11$\times {10}^{-4}$)
MW2	IGD	2.4671$\times {10}^{-2}$ (3.02$\times {10}^{-3}$)−	1.2472$\times {10}^{-2}$ (7.71$\times {10}^{-3}$)≈	2.0246$\times {10}^{-2}$ (7.25$\times {10}^{-3}$)≈	2.4547$\times {10}^{-2}$ (1.05$\times {10}^{-2}$)≈	1.6666$\times {10}^{-2}$ (5.24$\times {10}^{-3}$)≈	9.9745$\times {10}^{-2}$ (7.75$\times {10}^{-2}$)−	9.7542$\times {10}^{-3}$ (6.86$\times {10}^{-3}$)
	HV	5.4660$\times {10}^{-1}$ (4.52$\times {10}^{-3}$)−	5.7204$\times {10}^{-1}$ (1.35$\times {10}^{-2}$)≈	5.5418$\times {10}^{-1}$ (1.22$\times {10}^{-2}$)≈	5.4719$\times {10}^{-1}$ (1.53$\times {10}^{-2}$)≈	5.5961$\times {10}^{-1}$ (9.09$\times {10}^{-3}$)≈	4.5061$\times {10}^{-1}$ (9.76$\times {10}^{-2}$)−	5.6692$\times {10}^{-1}$ (1.23$\times {10}^{-2}$)
MW3	IGD	6.0297$\times {10}^{-3}$ (3.18$\times {10}^{-4}$)−	5.5115$\times {10}^{-3}$ (3.80$\times {10}^{-4}$)≈	5.0220$\times {10}^{-3}$ (2.42$\times {10}^{-4}$)≈	5.0630$\times {10}^{-3}$ (2.22$\times {10}^{-4}$)≈	5.7086$\times {10}^{-3}$ (6.79$\times {10}^{-4}$)≈	5.7481$\times {10}^{-3}$ (6.44$\times {10}^{-4}$)−	4.9940$\times {10}^{-3}$ (1.77$\times {10}^{-4}$)
	HV	5.4301$\times {10}^{-1}$ (4.90$\times {10}^{-4}$)−	5.4384$\times {10}^{-1}$ (6.64$\times {10}^{-4}$)≈	5.4414$\times {10}^{-1}$ (4.63$\times {10}^{-4}$)≈	5.4420$\times {10}^{-1}$ (2.42$\times {10}^{-4}$)≈	5.4380$\times {10}^{-1}$ (9.08$\times {10}^{-4}$)≈	5.4461$\times {10}^{-1}$ (2.96$\times {10}^{-4}$)≈	5.4423$\times {10}^{-1}$ (3.89$\times {10}^{-4}$)
MW4	IGD	6.5117$\times {10}^{-3}$ (4.68$\times {10}^{-4}$)≈	6.8851$\times {10}^{-3}$ (8.38$\times {10}^{-4}$)≈	6.3658$\times {10}^{-3}$ (1.49$\times {10}^{-4}$)≈	6.2140$\times {10}^{-3}$ (1.04$\times {10}^{-4}$)≈	1.3715$\times {10}^{-2}$ (1.72$\times {10}^{-2}$)≈	1.7708$\times {10}^{-2}$ (2.16$\times {10}^{-2}$)≈	5.9330$\times {10}^{-3}$ (8.21$\times {10}^{-4}$)
	HV	5.7905$\times {10}^{-1}$ (5.68$\times {10}^{-4}$)≈	5.7869$\times {10}^{-1}$ (9.96$\times {10}^{-4}$)≈	5.7923$\times {10}^{-1}$ (1.83$\times {10}^{-4}$)≈	5.7941$\times {10}^{-1}$ (1.06$\times {10}^{-3}$)≈	5.7114$\times {10}^{-1}$ (1.93$\times {10}^{-2}$)≈	5.6173$\times {10}^{-1}$ (3.32$\times {10}^{-2}$)≈	5.7969$\times {10}^{-1}$ (1.25$\times {10}^{-4}$)
MW5	IGD	2.6810$\times {10}^{-2}$ (2.40$\times {10}^{-2}$)≈	1.9416$\times {10}^{-2}$ (4.25$\times {10}^{-3}$)−	2.3415$\times {10}^{-3}$ (2.03$\times {10}^{-3}$)≈	6.6335$\times {10}^{-4}$ (2.03$\times {10}^{-4}$)+	2.7408$\times {10}^{-1}$ (3.44$\times {10}^{-1}$)−	5.7111$\times {10}^{-1}$ (3.67$\times {10}^{-1}$)−	3.3726$\times {10}^{-3}$ (3.73$\times {10}^{-3}$)
	HV	3.0361$\times {10}^{-1}$ (2.44$\times {10}^{-2}$)≈	3.1258$\times {10}^{-1}$ (2.43$\times {10}^{-3}$)−	3.2358$\times {10}^{-1}$ (9.60$\times {10}^{-4}$)≈	3.2429$\times {10}^{-1}$ (1.20$\times {10}^{-4}$)+	2.1870$\times {10}^{-1}$ (1.14$\times {10}^{-1}$)−	1.4690$\times {10}^{-1}$ (1.12$\times {10}^{-1}$)−	3.2318$\times {10}^{-1}$ (1.23$\times {10}^{-3}$)
MW6	IGD	2.0907$\times {10}^{-2}$ (1.12$\times {10}^{-2}$)−	1.1971$\times {10}^{-2}$ (3.99$\times {10}^{-3}$)−	2.8164$\times {10}^{-2}$ (7.15$\times {10}^{-3}$)−	1.3571$\times {10}^{-2}$ (7.17$\times {10}^{-3}$)≈	1.2743$\times {10}^{-1}$ (2.11$\times {10}^{-1}$)≈	2.8905$\times {10}^{-1}$ (2.99$\times {10}^{-1}$)−	4.0866$\times {10}^{-3}$ (2.67$\times {10}^{-3}$)
	HV	3.0148$\times {10}^{-1}$ (1.53$\times {10}^{-2}$)−	3.1035$\times {10}^{-1}$ (5.24$\times {10}^{-3}$)−	2.9163$\times {10}^{-1}$ (9.69$\times {10}^{-3}$)−	3.1140$\times {10}^{-1}$ (9.73$\times {10}^{-3}$)≈	2.7096$\times {10}^{-1}$ (6.32$\times {10}^{-2}$)≈	1.8611$\times {10}^{-1}$ (1.19$\times {10}^{-1}$)−	3.2608$\times {10}^{-1}$ (4.81$\times {10}^{-3}$)
MW7	IGD	5.3408$\times {10}^{-3}$ (4.65$\times {10}^{-5}$)≈	7.4288$\times {10}^{-3}$ (8.46$\times {10}^{-4}$)−	5.0896$\times {10}^{-3}$ (4.13$\times {10}^{-4}$)≈	4.7052$\times {10}^{-3}$ (4.41$\times {10}^{-4}$)≈	5.3385$\times {10}^{-3}$ (2.61$\times {10}^{-4}$)≈	9.4991$\times {10}^{-3}$ (1.77$\times {10}^{-3}$)−	5.2352$\times {10}^{-3}$ (6.35$\times {10}^{-4}$)
	HV	4.1103$\times {10}^{-1}$ (4.54$\times {10}^{-4}$)≈	4.0898$\times {10}^{-1}$ (1.15$\times {10}^{-3}$)−	4.1163$\times {10}^{-1}$ (5.70$\times {10}^{-4}$)≈	4.1196$\times {10}^{-1}$ (2.54$\times {10}^{-4}$)≈	4.1143$\times {10}^{-1}$ (5.30$\times {10}^{-4}$)≈	4.1257$\times {10}^{-1}$ (2.20$\times {10}^{-4}$)≈	4.1174$\times {10}^{-1}$ (1.04$\times {10}^{-3}$)
MW8	IGD	2.9668$\times {10}^{-2}$ (1.86$\times {10}^{-2}$)≈	9.4994$\times {10}^{-3}$ (7.90$\times {10}^{-3}$)≈	2.0580$\times {10}^{-2}$ (1.22$\times {10}^{-2}$)≈	2.5545$\times {10}^{-2}$ (1.07$\times {10}^{-2}$)≈	2.3854$\times {10}^{-2}$ (7.38$\times {10}^{-3}$)≈	4.9277$\times {10}^{-2}$ (1.91$\times {10}^{-2}$)−	2.1166$\times {10}^{-2}$ (7.10$\times {10}^{-3}$)
	HV	2.8146$\times {10}^{-1}$ (2.86$\times {10}^{-2}$)≈	3.1200$\times {10}^{-1}$ (1.25$\times {10}^{-2}$)≈	2.9535$\times {10}^{-1}$ (1.85$\times {10}^{-2}$)≈	2.8781$\times {10}^{-1}$ (1.61$\times {10}^{-2}$)≈	2.9040$\times {10}^{-1}$ (1.11$\times {10}^{-2}$)≈	2.5114$\times {10}^{-1}$ (3.01$\times {10}^{-2}$)−	2.9445$\times {10}^{-1}$ (1.07$\times {10}^{-2}$)
MW9	IGD	9.5797$\times {10}^{-3}$ (2.31$\times {10}^{-3}$)−	8.8084$\times {10}^{-3}$ (6.27$\times {10}^{-4}$)−	4.8282$\times {10}^{-3}$ (1.99$\times {10}^{-4}$)≈	5.5030$\times {10}^{-3}$ (6.19$\times {10}^{-4}$)≈	6.9900$\times {10}^{-3}$ (1.14$\times {10}^{-3}$)−	5.3824$\times {10}^{-3}$ (9.78$\times {10}^{-4}$)≈	4.9346$\times {10}^{-3}$ (2.59$\times {10}^{-4}$)
	HV	3.8496$\times {10}^{-1}$ (5.05$\times {10}^{-3}$)−	3.9184$\times {10}^{-1}$ (6.16$\times {10}^{-4}$)−	3.9794$\times {10}^{-1}$ (7.56$\times {10}^{-4}$)≈	3.9400$\times {10}^{-1}$ (2.72$\times {10}^{-3}$)≈	3.9039$\times {10}^{-1}$ (2.34$\times {10}^{-3}$)−	3.9463$\times {10}^{-1}$ (3.56$\times {10}^{-3}$)≈	3.9903$\times {10}^{-1}$ (2.90$\times {10}^{-3}$)
MW10	IGD	1.2505$\times {10}^{-1}$ (1.00$\times {10}^{-1}$)≈	1.1542$\times {10}^{-2}$ (7.19$\times {10}^{-3}$)≈	1.6410$\times {10}^{-2}$ (9.40$\times {10}^{-3}$)≈	3.2628$\times {10}^{-2}$ (1.69$\times {10}^{-2}$)≈	6.4999$\times {10}^{-2}$ (5.80$\times {10}^{-2}$)≈	3.4954$\times {10}^{-1}$ (2.42$\times {10}^{-1}$)≈	3.5918$\times {10}^{-2}$ (2.11$\times {10}^{-2}$)
	HV	3.6196$\times {10}^{-1}$ (5.67$\times {10}^{-2}$)≈	4.4202$\times {10}^{-1}$ (9.72$\times {10}^{-3}$)≈	4.3581$\times {10}^{-1}$ (9.59$\times {10}^{-3}$)≈	4.2120$\times {10}^{-1}$ (1.37$\times {10}^{-2}$)≈	3.9905$\times {10}^{-1}$ (3.74$\times {10}^{-2}$)≈	2.4603$\times {10}^{-1}$ (1.12$\times {10}^{-1}$)≈	4.1867$\times {10}^{-1}$ (1.65$\times {10}^{-2}$)
MW11	IGD	4.3943$\times {10}^{-1}$ (3.27$\times {10}^{-1}$)−	1.5706$\times {10}^{-2}$ (3.92$\times {10}^{-4}$)−	6.4006$\times {10}^{-3}$ (3.57$\times {10}^{-4}$)≈	6.2673$\times {10}^{-3}$ (1.31$\times {10}^{-4}$)≈	6.1445$\times {10}^{-3}$ (2.50$\times {10}^{-4}$)≈	7.5658$\times {10}^{-3}$ (2.94$\times {10}^{-4}$)≈	6.1887$\times {10}^{-3}$ (9.15$\times {10}^{-5}$)
	HV	3.3014$\times {10}^{-1}$ (8.08$\times {10}^{-2}$)−	4.4209$\times {10}^{-1}$ (6.97$\times {10}^{-4}$)−	4.4698$\times {10}^{-1}$ (2.75$\times {10}^{-4}$)≈	4.4729$\times {10}^{-1}$ (1.01$\times {10}^{-4}$)≈	4.4726$\times {10}^{-1}$ (4.86$\times {10}^{-4}$)≈	4.4640$\times {10}^{-1}$ (3.33$\times {10}^{-4}$)≈	4.4701$\times {10}^{-1}$ (1.41$\times {10}^{-4}$)
MW12	IGD	3.8931$\times {10}^{-1}$ (4.43$\times {10}^{-1}$)−	7.9188$\times {10}^{-3}$ (6.00$\times {10}^{-4}$)−	4.9123$\times {10}^{-3}$ (4.20$\times {10}^{-5}$)≈	5.2133$\times {10}^{-3}$ (4.08$\times {10}^{-4}$)≈	5.6206$\times {10}^{-3}$ (1.47$\times {10}^{-3}$)≈	5.0237$\times {10}^{-3}$ (8.91$\times {10}^{-5}$)≈	4.9434$\times {10}^{-3}$ (1.59$\times {10}^{-4}$)
	HV	3.0191$\times {10}^{-1}$ (3.49$\times {10}^{-1}$)−	6.0017$\times {10}^{-1}$ (3.37$\times {10}^{-4}$)−	6.0427$\times {10}^{-1}$ (1.67$\times {10}^{-4}$)≈	6.0389$\times {10}^{-1}$ (1.04$\times {10}^{-3}$)≈	6.0343$\times {10}^{-1}$ (2.24$\times {10}^{-3}$)≈	6.0437$\times {10}^{-1}$ (3.54$\times {10}^{-4}$)≈	6.0430$\times {10}^{-1}$ (2.83$\times {10}^{-4}$)
MW13	IGD	2.2901$\times {10}^{-1}$ (2.21$\times {10}^{-1}$)−	3.9569$\times {10}^{-2}$ (3.57$\times {10}^{-2}$)≈	6.6127$\times {10}^{-2}$ (4.23$\times {10}^{-2}$)≈	9.5446$\times {10}^{-2}$ (3.38$\times {10}^{-2}$)−	1.9761$\times {10}^{-1}$ (5.80$\times {10}^{-2}$)−	1.7622$\times {10}^{-1}$ (2.19$\times {10}^{-2}$)−	2.9415$\times {10}^{-2}$ (1.34$\times {10}^{-2}$)
	HV	4.0406$\times {10}^{-1}$ (2.86$\times {10}^{-2}$)−	4.6042$\times {10}^{-1}$ (1.59$\times {10}^{-2}$)≈	4.4585$\times {10}^{-1}$ (2.24$\times {10}^{-2}$)≈	4.3144$\times {10}^{-1}$ (2.03$\times {10}^{-2}$)−	3.8627$\times {10}^{-1}$ (1.81$\times {10}^{-2}$)−	4.0737$\times {10}^{-1}$ (1.89$\times {10}^{-2}$)−	4.6281$\times {10}^{-1}$ (4.63$\times {10}^{-3}$)
MW14	IGD	1.9127$\times {10}^{-2}$ (2.02$\times {10}^{-3}$)≈	4.6069$\times {10}^{-1}$ (4.55$\times {10}^{-1}$)−	1.6658$\times {10}^{-2}$ (2.18$\times {10}^{-3}$)≈	1.6059$\times {10}^{-2}$ (7.25$\times {10}^{-4}$)≈	1.7623$\times {10}^{-2}$ (3.00$\times {10}^{-3}$)≈	1.4583$\times {10}^{-2}$ (3.18$\times {10}^{-4}$)≈	1.7480$\times {10}^{-2}$ (5.67$\times {10}^{-3}$)
	HV	5.0152$\times {10}^{-1}$ (1.47$\times {10}^{-3}$)≈	4.1331$\times {10}^{-1}$ (1.03$\times {10}^{-1}$)−	5.0356$\times {10}^{-1}$ (1.66$\times {10}^{-3}$)≈	5.0432$\times {10}^{-1}$ (6.52$\times {10}^{-4}$)≈	5.0269$\times {10}^{-1}$ (2.09$\times {10}^{-3}$)≈	5.0507$\times {10}^{-1}$ (1.66$\times {10}^{-4}$)≈	5.0293$\times {10}^{-1}$ (3.86$\times {10}^{-3}$)
$+/-/\approx$	0/15/13	0/14/14	2/26/00	2/24/02	6/22/00	0/13/15

* Data with a dark background shows that the algorithm has reached the optimal value for the relevant problem index.

. Across all 28 test cases, DDCRO achieved 10 optimal results, while CAEAD and CMODE-FTR each obtained five optimal results. URCMO, CCMO, and TSTI secured four, three, and one optimal results, respectively, whereas NSGA-II failed to obtain any optimal results. To more clearly observe the differences in the optimal populations obtained by each algorithm, Figure 3 illustrates the population distributions of each algorithm on the MW13 problem. This problem is characterized by infeasible-region barriers, a complex feasible-region shape, and a discontinuous CPF, which poses high demands on the algorithms’ search capabilities. As depicted in the figure, CAEAD employs a dynamic evolutionary/degenerative strategy for its auxiliary population, which aids in rapid convergence. However, DDCRO’s dual-stage strategy, combined with its weak complementary mechanism, allows it to maintain better population diversity while still achieving rapid convergence. This is evident in problems with complex feasible-region shapes like MW13. Compared with DDCRO, NSGA-II results in inadequate population convergence due to its excessive focus on constraint satisfaction, and URCMO may fail to effectively learn the problem type during the first phase, which ultimately results in an imbalance of population convergence and diversity. Observing URCMO and DDCRO, URCMO learns the problem types in the first stage and customizes the evolution strategy for the auxiliary population accordingly. However, DDCRO provides a more robust solution, especially in the scene with discontinuous CPF, through the dual-population mechanism and the structured method of directional search operators to deal with constraints.

4.2.3. Comparisons on DOC

Considering the narrow feasible region characteristics of the DOC test suite under dual constraints in both decision and objective spaces, this study verifies the algorithms’ convergence capabilities within extremely constrained spaces through 18 test cases.

Table 4. IGD values and HV values obtained by seven algorithms on DOC test suite.

Problem		NSGAII	CAEAD	CCMO	URCMO	TSTI	CMODE-FTR	DDCRO
DOC1	IGD	2.9253$\times {10}^0$ (3.34$\times {10}^0$)−	8.2729$\times {10}^2$ (2.68$\times {10}^2$)−	1.0472$\times {10}^1$ (4.00$\times {10}^0$)−	1.1683$\times {10}^{-1}$ (2.20$\times {10}^{-1}$)+	3.8109$\times {10}^0$ (4.12$\times {10}^0$)−	1.1068$\times {10}^1$ (8.70$\times {10}^0$)−	7.4209$\times {10}^{-1}$ (1.64$\times {10}^{-1}$1)
	HV	1.7695$\times {10}^{-3}$ (3.54$\times {10}^{-3}$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	2.9030$\times {10}^{-1}$ (1.03$\times {10}^{-1}$)+	6.7356$\times {10}^{-2}$ (1.35$\times {10}^{-1}$)≈	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	9.0909$\times {10}^{-2}$ (4.01$\times {10}^{-1}$1)
DOC2	IGD	NaN (NaN)	1.1290$\times {10}^{-2}$ (1.48$\times {10}^{-2}$)−	NaN (NaN)	3.9553$\times {10}^{-1}$ (9.18$\times {10}^{-2}$)−	NaN (NaN)	3.4578$\times {10}^{-3}$ (6.94$\times {10}^{-5}$)−	2.1205$\times {10}^{-3}$ (5.25$\times {10}^{-4}$)
	HV	NaN (NaN)	6.1105$\times {10}^{-1}$ (2.05$\times {10}^{-2}$)−	NaN (NaN)	3.7072$\times {10}^{-1}$ (6.10$\times {10}^{-2}$)≈	NaN (NaN)	6.1145$\times {10}^{-1}$ (9.26$\times {10}^{-5}$)−	6.2079$\times {10}^{-1}$ (9.50$\times {10}^{-4}$)
DOC3	IGD	6.0806$\times {10}^2$ (2.43$\times {10}^2$)−	2.9889$\times {10}^{-1}$ (5.62$\times {10}^{-1}$)≈	7.8636$\times {10}^2$ (0.00$\times {10}^0$)−	1.0740$\times {10}^2$ (1.95$\times {10}^2$)−	6.4069$\times {10}^2$ (2.39$\times {10}^2$)−	7.3797$\times {10}^{-1}$ (5.30$\times {10}^{-3}$)−	2.5982$\times {10}^{-1}$ (3.42$\times {10}^{-1}$)
	HV	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	2.1173$\times {10}^{-1}$ (1.61$\times {10}^{-1}$)≈	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	1.0533$\times {10}^{-1}$ (1.53$\times {10}^{-2}$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	1.3189$\times {10}^{-1}$ (6.35$\times {10}^{-3}$)−	2.2318$\times {10}^{-1}$ (1.21$\times {10}^{-1}$)
DOC4	IGD	1.4097$\times {10}^0$ (1.58$\times {10}^0$)−	3.4608$\times {10}^2$ (1.24$\times {10}^2$)−	1.6812$\times {10}^0$ (2.13$\times {10}^0$)−	4.0835$\times {10}^{-2}$ (9.57$\times {10}^{-3}$)+	2.6397$\times {10}^0$ (2.81$\times {10}^0$)−	1.0652$\times {10}^0$ (6.03$\times {10}^{-1}$)≈	7.8113$\times {10}^{-1}$ (1.31$\times {10}^{-1}$4)
	HV	4.7278$\times {10}^{-2}$ (6.25$\times {10}^{-2}$)+	5.4256$\times {10}^{-3}$ (3.12$\times {10}^{-3}$)−	8.4613$\times {10}^{-2}$ (1.07$\times {10}^{-1}$)≈	5.1678$\times {10}^{-1}$ (1.09$\times {10}^{-2}$)+	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	4.2122$\times {10}^{-2}$ (8.42$\times {10}^{-2}$)≈	2.8087$\times {10}^{-3}$ (1.47$\times {10}^{-1}$4)
DOC5	IGD	NaN (NaN)	2.3946$\times {10}^{-2}$ (8.44$\times {10}^{-4}$)+	NaN (NaN)	1.3060$\times {10}^2$ (3.41$\times {10}^{-2}$)≈	8.1993$\times {10}^1$ (3.33$\times {10}^1$)≈	9.5431$\times {10}^1$ (7.52$\times {10}^1$)−	8.7037$\times {10}^1$ (7.53$\times {10}^1$)
	HV	NaN (NaN)	1.3613$\times {10}^{-1}$ (8.62$\times {10}^{-4}$)−	NaN (NaN)	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	NaN (NaN)	1.5933$\times {10}^{-1}$ (2.76$\times {10}^{-1}$)
DOC6	IGD	1.6480$\times {10}^0$ (2.37$\times {10}^0$)−	3.0718$\times {10}^1$ (1.04$\times {10}^1$)−	2.8738$\times {10}^0$ (2.12$\times {10}^0$)−	3.9294$\times {10}^{-3}$ (8.91$\times {10}^{-4}$)≈	1.4032$\times {10}^0$ (8.81$\times {10}^{-1}$)−	1.8493$\times {10}^0$ (1.54$\times {10}^0$)−	2.9024$\times {10}^{-3}$ (3.94$\times {10}^{-4}$)
	HV	6.6140$\times {10}^{-2}$ (8.60$\times {10}^{-2}$)≈	5.3306$\times {10}^{-1}$ (3.43$\times {10}^{-3}$)≈	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	4.9828$\times {10}^{-1}$ (2.02$\times {10}^{-2}$)≈	2.9290$\times {10}^{-2}$ (5.86$\times {10}^{-2}$)≈	9.9672$\times {10}^{-2}$ (1.99$\times {10}^{-1}$)−	5.3333$\times {10}^{-1}$ (8.20$\times {10}^{-3}$)
DOC7	IGD	5.7742$\times {10}^0$ (2.22$\times {10}^0$)−	3.7170$\times {10}^{-2}$ (6.79$\times {10}^{-2}$)−	5.2305$\times {10}^0$ (1.52$\times {10}^0$)−	6.4552$\times {10}^{-3}$ (3.02$\times {10}^{-3}$)−	6.6391$\times {10}^0$ (1.82$\times {10}^0$)−	5.9332$\times {10}^0$ (9.99$\times {10}^{-1}$)−	2.3762$\times {10}^{-3}$ (1.06$\times {10}^{-4}$)
	HV	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	5.3605$\times {10}^{-1}$ (2.52$\times {10}^{-3}$)+	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	4.6720$\times {10}^{-1}$ (5.50$\times {10}^{-2}$)≈	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	4.4292$\times {10}^{-1}$ (1.50$\times {10}^{-1}$)
DOC8	IGD	8.0012$\times {10}^1$ (7.68$\times {10}^1$)−	6.2107$\times {10}^{-2}$ (1.93$\times {10}^{-3}$)+	1.1374$\times {10}^2$ (4.45$\times {10}^1$)−	4.0650$\times {10}^0$ (7.61$\times {10}^{-1}$)−	1.2127$\times {10}^2$ (5.85$\times {10}^1$)−	1.3146$\times {10}^2$ (6.82$\times {10}^1$)−	2.6482$\times {10}^{-1}$ (9.77$\times {10}^{-2}$)
	HV	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	8.0093$\times {10}^{-1}$ (1.95$\times {10}^{-3}$)+	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	5.2764$\times {10}^{-1}$ (1.21$\times {10}^{-1}$)
DOC9	IGD	1.9469$\times {10}^{-1}$ (8.49$\times {10}^{-2}$)≈	9.2808$\times {10}^{-2}$ (9.97$\times {10}^{-3}$)≈	2.0132$\times {10}^{-1}$ (1.11$\times {10}^{-1}$)≈	9.2210$\times {10}^{-2}$ (1.58$\times {10}^{-2}$)≈	1.1037$\times {10}^{-1}$ (1.73$\times {10}^{-1}$)≈	2.8189$\times {10}^{-1}$ (1.53$\times {10}^{-2}$)−	7.6204$\times {10}^{-2}$ (9.33$\times {10}^{-3}$)
	HV	0.0000$\times {10}^0$ (0.00$\times {10}^0$)+	NaN (NaN)	NaN (NaN)	NaN (NaN)	NaN (NaN)	NaN (NaN)	NaN (NaN)
$+/-/\approx$	2/14/2	5/9/04	0/16/2	8/6/04	0/14/4	0/16/2

* Data with a dark background shows that the algorithm has reached the optimal value for the relevant problem index.

presents the experimental results of each algorithm on this problem. Across all test cases, DDCRO achieved nine optimal results, while CAEAD and URCMO each obtained four optimal results. NSGA-II secured one optimal result, and the remaining algorithms failed to obtain any optimal results. The poor performance of CCMO stems from its auxiliary population’s continuous search for the UPF, which fails to provide information for locating the CPF. TSTI also proves ineffective for solving problems with narrow feasible regions due to relying solely on a phased selection strategy. Similarly, despite dynamically adjusting its selection strategy, CMODE-FTR cannot fully leverage its advantages. URCMO also adopts a phased optimization strategy, which plays a vital role in solving these problems. In contrast, DDCRO’s refined molecular collision operators and structure repair strategy ensure that high-quality solutions are preserved during the evolutionary process. This leads to better performance on problems with extremely narrow feasible regions. To more clearly observe the performance differences among the algorithms, Figure 4 provides the population distribution of each algorithm on the DOC6 problem. It can be observed that the populations of NSGA-II, CCMO, TSTI, and CMODE-FTR all become trapped in local optima. Although CAEAD’s population can search near the feasible region using a degenerative mechanism, it still fails to guarantee population diversity. In comparison, the populations of URCMO and DDCRO are better distributed along the CPF. Due to the global search strategy in the first stage, DDCRO exhibits stronger search capabilities and better population distribution on discontinuous CPF segments.

4.2.4. Comparisons on LIRCMOP

To validate the robustness of the algorithm in scenarios with extreme infeasible regions and mixed-type CPFs, this experiment selected 28 high-dimensional test problems from the LIR-CMOP series, which focuses on examining the algorithm’s adaptability to core challenges such as fragmented feasible regions and spatial dislocation between the UPF and CPF.

Table 5. IGD values and HV values obtained by seven algorithms on LIRCMOP test suite.

Problem		NSGAII	CAEAD	CCMO	URCMO	TSTI	CMODE-FTR	DDCRO
LIRCMOP1	IGD	2.7449$\times {10}^{-1}$ (2.83$\times {10}^{-2}$)−	4.3540$\times {10}^{-1}$ (7.39$\times {10}^{-2}$)−	2.7039$\times {10}^{-1}$ (6.07$\times {10}^{-2}$)−	9.9904$\times {10}^{-2}$ (6.64$\times {10}^{-2}$)≈	2.1876$\times {10}^{-1}$ (1.96$\times {10}^{-2}$)≈	2.6361$\times {10}^{-1}$ (4.14$\times {10}^{-2}$)−	1.6329$\times {10}^{-2}$ (5.52$\times {10}^{-3}$)
	HV	1.1980$\times {10}^{-1}$ (4.59$\times {10}^{-3}$)≈	1.0044$\times {10}^{-1}$ (4.03$\times {10}^{-3}$)−	1.1732$\times {10}^{-1}$ (2.06$\times {10}^{-2}$)−	1.9313$\times {10}^{-1}$ (3.11$\times {10}^{-2}$)≈	1.4142$\times {10}^{-1}$ (3.46$\times {10}^{-3}$)≈	1.1921$\times {10}^{-1}$ (1.08$\times {10}^{-2}$)−	2.3278$\times {10}^{-1}$ (3.89$\times {10}^{-3}$)
LIRCMOP2	IGD	2.5970$\times {10}^{-1}$ (3.35$\times {10}^{-2}$)−	1.8335$\times {10}^{-1}$ (4.95$\times {10}^{-2}$)−	2.0220$\times {10}^{-1}$ (7.73$\times {10}^{-2}$)−	4.1306$\times {10}^{-3}$ (1.69$\times {10}^{-4}$)+	1.8454$\times {10}^{-1}$ (2.22$\times {10}^{-2}$)−	2.2267$\times {10}^{-1}$ (4.32$\times {10}^{-2}$)−	1.2727$\times {10}^{-2}$ (1.53$\times {10}^{-3}$)
	HV	2.2756$\times {10}^{-1}$ (1.45$\times {10}^{-2}$)−	2.7582$\times {10}^{-1}$ (4.05$\times {10}^{-2}$)−	2.6006$\times {10}^{-1}$ (4.52$\times {10}^{-2}$)−	3.6091$\times {10}^{-1}$ (1.97$\times {10}^{-4}$)+	2.6764$\times {10}^{-1}$ (1.01$\times {10}^{-2}$)≈	2.4536$\times {10}^{-1}$ (2.33$\times {10}^{-2}$)−	3.5653$\times {10}^{-1}$ (5.87$\times {10}^{-4}$)
LIRCMOP3	IGD	3.1570$\times {10}^{-1}$ (2.76$\times {10}^{-2}$)−	2.9752$\times {10}^{-1}$ (6.46$\times {10}^{-2}$)≈	2.5325$\times {10}^{-1}$ (6.05$\times {10}^{-2}$)−	9.2023$\times {10}^{-2}$ (6.54$\times {10}^{-2}$)−	2.2847$\times {10}^{-1}$ (2.86$\times {10}^{-2}$)≈	2.4553$\times {10}^{-1}$ (6.84$\times {10}^{-2}$)−	1.6157$\times {10}^{-2}$ (8.94$\times {10}^{-3}$)
	HV	9.7716$\times {10}^{-2}$ (7.23$\times {10}^{-3}$)−	9.9980$\times {10}^{-2}$ (1.86$\times {10}^{-2}$)−	1.1811$\times {10}^{-1}$ (1.77$\times {10}^{-2}$)≈	1.6650$\times {10}^{-1}$ (2.97$\times {10}^{-2}$)−	1.2211$\times {10}^{-1}$ (5.85$\times {10}^{-3}$)−	1.2205$\times {10}^{-1}$ (2.22$\times {10}^{-2}$)−	2.0276$\times {10}^{-1}$ (3.66$\times {10}^{-3}$)
LIRCMOP4	IGD	2.8082$\times {10}^{-1}$ (3.00$\times {10}^{-2}$)−	3.0014$\times {10}^{-1}$ (4.86$\times {10}^{-2}$)−	2.4722$\times {10}^{-1}$ (6.36$\times {10}^{-2}$)−	1.0687$\times {10}^{-1}$ (8.14$\times {10}^{-2}$)−	2.0825$\times {10}^{-1}$ (1.61$\times {10}^{-2}$)−	2.0094$\times {10}^{-1}$ (4.89$\times {10}^{-2}$)−	7.8443$\times {10}^{-3}$ (1.80$\times {10}^{-3}$)
	HV	1.9400$\times {10}^{-1}$ (1.33$\times {10}^{-2}$)−	1.8673$\times {10}^{-1}$ (2.62$\times {10}^{-2}$)−	2.0858$\times {10}^{-1}$ (3.30$\times {10}^{-2}$)−	2.6692$\times {10}^{-1}$ (3.57$\times {10}^{-2}$)−	2.2891$\times {10}^{-1}$ (1.65$\times {10}^{-2}$)≈	2.3137$\times {10}^{-1}$ (2.28$\times {10}^{-2}$)−	3.1304$\times {10}^{-1}$ (1.09$\times {10}^{-3}$)
LIRCMOP5	IGD	1.2185$\times {10}^0$ (5.52$\times {10}^{-3}$)−	1.2163$\times {10}^0$ (1.40$\times {10}^{-2}$)−	1.0537$\times {10}^{-2}$ (2.45$\times {10}^{-3}$)−	3.3111$\times {10}^{-1}$ (1.83$\times {10}^{-2}$)−	7.6531$\times {10}^{-1}$ (5.21$\times {10}^{-1}$)−	2.7959$\times {10}^{-1}$ (2.68$\times {10}^{-2}$)−	7.6998$\times {10}^{-3}$ (2.42$\times {10}^{-4}$)
	HV	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	1.4138$\times {10}^{-1}$ (7.84$\times {10}^{-3}$)−	2.8917$\times {10}^{-1}$ (1.78$\times {10}^{-3}$)≈	7.6955$\times {10}^{-2}$ (8.97$\times {10}^{-2}$)−	1.6260$\times {10}^{-1}$ (1.16$\times {10}^{-2}$)−	2.9072$\times {10}^{-1}$ (1.98$\times {10}^{-4}$)
LIRCMOP6	IGD	1.1048$\times {10}^0$ (4.82$\times {10}^{-1}$)−	1.3463$\times {10}^0$ (1.19$\times {10}^{-3}$)−	9.1878$\times {10}^{-3}$ (8.96$\times {10}^{-5}$)−	2.3687$\times {10}^{-1}$ (3.46$\times {10}^{-2}$)−	1.0955$\times {10}^0$ (5.00$\times {10}^{-1}$)−	3.2387$\times {10}^{-1}$ (2.63$\times {10}^{-2}$)−	7.0283$\times {10}^{-3}$ (7.48$\times {10}^{-4}$)
	HV	2.5329$\times {10}^{-2}$ (5.07$\times {10}^{-2}$)−	0.0000$\times {10}^0$ (0.00$\times {10}^0$)−	1.2749$\times {10}^{-1}$ (1.33$\times {10}^{-2}$)−	1.9506$\times {10}^{-1}$ (9.64$\times {10}^{-5}$)≈	2.5013$\times {10}^{-2}$ (5.00$\times {10}^{-2}$)−	1.0974$\times {10}^{-1}$ (6.80$\times {10}^{-3}$)−	1.9652$\times {10}^{-1}$ (1.87$\times {10}^{-4}$)
LIRCMOP7	IGD	1.6731$\times {10}^{-1}$ (5.22$\times {10}^{-2}$)−	1.8871$\times {10}^{-2}$ (1.06$\times {10}^{-1}$)+	1.0954$\times {10}^{-1}$ (2.30$\times {10}^{-2}$)−	7.6530$\times {10}^{-3}$ (2.92$\times {10}^{-4}$)+	1.3837$\times {10}^{-1}$ (3.18$\times {10}^{-2}$)−	1.0084$\times {10}^{-1}$ (1.34$\times {10}^{-2}$)−	8.2893$\times {10}^{-2}$ (2.79$\times {10}^{-4}$)
	HV	2.3458$\times {10}^{-1}$ (1.45$\times {10}^{-2}$)−	2.9195$\times {10}^{-1}$ (2.99$\times {10}^{-2}$)≈	2.5135$\times {10}^{-1}$ (6.52$\times {10}^{-3}$)≈	2.9424$\times {10}^{-1}$ (1.66$\times {10}^{-4}$)≈	2.4317$\times {10}^{-1}$ (1.01$\times {10}^{-2}$)≈	2.5397$\times {10}^{-1}$ (2.77$\times {10}^{-3}$)−	2.9406$\times {10}^{-1}$ (1.49$\times {10}^{-4}$)
LIRCMOP8	IGD	1.3234$\times {10}^0$ (7.17$\times {10}^{-1}$)−	2.6366$\times {10}^{-2}$ (5.27$\times {10}^{-2}$)+	1.5128$\times {10}^{-1}$ (3.01$\times {10}^{-2}$)−	7.6717$\times {10}^{-3}$ (9.77$\times {10}^{-5}$)+	2.3944$\times {10}^{-1}$ (1.03$\times {10}^{-1}$)−	1.2381$\times {10}^{-1}$ (3.59$\times {10}^{-2}$)−	8.3020$\times {10}^{-2}$ (1.25$\times {10}^{-4}$)
	HV	5.7138$\times {10}^{-2}$ (1.14$\times {10}^{-1}$)−	2.9386$\times {10}^{-1}$ (8.89$\times {10}^{-3}$)≈	2.4035$\times {10}^{-1}$ (9.93$\times {10}^{-3}$)≈	2.9422$\times {10}^{-1}$ (5.42$\times {10}^{-5}$)≈	2.2503$\times {10}^{-1}$ (9.15$\times {10}^{-3}$)≈	2.4606$\times {10}^{-1}$ (1.11$\times {10}^{-2}$)−	2.9407$\times {10}^{-1}$ (9.81$\times {10}^{-5}$)
LIRCMOP9	IGD	8.9381$\times {10}^{-1}$ (1.24$\times {10}^{-1}$)−	5.4657$\times {10}^{-1}$ (7.93$\times {10}^{-2}$)≈	3.7897$\times {10}^{-1}$ (7.71$\times {10}^{-2}$)−	2.0962$\times {10}^{-1}$ (2.02$\times {10}^{-1}$)≈	4.7970$\times {10}^{-1}$ (4.37$\times {10}^{-2}$)≈	3.9393$\times {10}^{-1}$ (1.02$\times {10}^{-1}$)−	1.0897$\times {10}^{-1}$ (5.71$\times {10}^{-2}$)
	HV	1.3587$\times {10}^{-1}$ (5.40$\times {10}^{-2}$)−	3.2977$\times {10}^{-1}$ (4.72$\times {10}^{-2}$)−	4.4378$\times {10}^{-1}$ (4.03$\times {10}^{-2}$)−	4.8093$\times {10}^{-1}$ (8.60$\times {10}^{-2}$)≈	3.7784$\times {10}^{-1}$ (2.28$\times {10}^{-2}$)≈	4.1103$\times {10}^{-1}$ (4.64$\times {10}^{-2}$)≈	5.2079$\times {10}^{-1}$ (4.55$\times {10}^{-2}$)
LIRCMOP10	IGD	8.3474$\times {10}^{-1}$ (1.17$\times {10}^{-3}$)−	2.5749$\times {10}^{-1}$ (1.12$\times {10}^{-1}$)−	9.8981$\times {10}^{-2}$ (2.06$\times {10}^{-2}$)≈	9.9064$\times {10}^{-3}$ (1.89$\times {10}^{-3}$)−	6.1928$\times {10}^{-1}$ (3.19$\times {10}^{-1}$)−	8.8364$\times {10}^{-2}$ (6.56$\times {10}^{-2}$)−	7.2569$\times {10}^{-3}$ (2.32$\times {10}^{-3}$)
	HV	1.1813$\times {10}^{-1}$ (9.21$\times {10}^{-4}$)−	5.6856$\times {10}^{-1}$ (5.56$\times {10}^{-2}$)−	6.6456$\times {10}^{-1}$ (4.22$\times {10}^{-3}$)−	7.0388$\times {10}^{-1}$ (9.51$\times {10}^{-4}$)≈	3.7660$\times {10}^{-1}$ (1.90$\times {10}^{-1}$)−	6.6076$\times {10}^{-1}$ (2.76$\times {10}^{-2}$)−	7.0565$\times {10}^{-1}$ (1.28$\times {10}^{-3}$)
LIRCMOP11	IGD	7.5204$\times {10}^{-1}$ (3.91$\times {10}^{-4}$)−	1.8885$\times {10}^{-1}$ (9.12$\times {10}^{-3}$)−	2.9826$\times {10}^{-2}$ (3.40$\times {10}^{-2}$)≈	4.2108$\times {10}^{-2}$ (3.51$\times {10}^{-2}$)−	4.3844$\times {10}^{-1}$ (1.79$\times {10}^{-1}$)−	5.0023$\times {10}^{-2}$ (3.17$\times {10}^{-2}$)≈	2.0630$\times {10}^{-2}$ (3.46$\times {10}^{-2}$)
	HV	2.2713$\times {10}^{-1}$ (2.64$\times {10}^{-4}$)−	6.1236$\times {10}^{-1}$ (3.72$\times {10}^{-3}$)−	6.7906$\times {10}^{-1}$ (1.72$\times {10}^{-2}$)≈	6.7366$\times {10}^{-1}$ (1.62$\times {10}^{-2}$)≈	4.4070$\times {10}^{-1}$ (8.59$\times {10}^{-2}$)−	6.6706$\times {10}^{-1}$ (1.36$\times {10}^{-2}$)≈	6.8511$\times {10}^{-1}$ (1.66$\times {10}^{-2}$)
LIRCMOP12	IGD	5.4822$\times {10}^{-1}$ (2.22$\times {10}^{-1}$)−	4.3325$\times {10}^{-1}$ (4.61$\times {10}^{-1}$)−	1.6727$\times {10}^{-1}$ (1.03$\times {10}^{-1}$)−	1.1282$\times {10}^{-1}$ (1.71$\times {10}^{-1}$)−	3.4805$\times {10}^{-1}$ (5.09$\times {10}^{-2}$)−	1.8868$\times {10}^{-1}$ (6.90$\times {10}^{-2}$)−	3.8137$\times {10}^{-3}$ (1.09$\times {10}^{-3}$)
	HV	3.4342$\times {10}^{-1}$ (1.05$\times {10}^{-1}$)−	4.2233$\times {10}^{-1}$ (1.98$\times {10}^{-1}$)−	5.4270$\times {10}^{-1}$ (5.70$\times {10}^{-2}$)−	5.6502$\times {10}^{-1}$ (8.76$\times {10}^{-2}$)−	4.4299$\times {10}^{-1}$ (2.82$\times {10}^{-2}$)−	5.3174$\times {10}^{-1}$ (3.58$\times {10}^{-2}$)−	6.2012$\times {10}^{-1}$ (2.65$\times {10}^{-4}$)
LIRCMOP13	IGD	1.3265$\times {10}^0$ (1.55$\times {10}^{-3}$)−	1.0806$\times {10}^{-1}$ (6.35$\times {10}^{-4}$)+	9.7572$\times {10}^{-2}$ (1.15$\times {10}^{-3}$)+	1.0247$\times {10}^{-1}$ (5.55$\times {10}^{-4}$)+	1.0091$\times {10}^0$ (6.13$\times {10}^{-1}$)≈	9.0807$\times {10}^{-2}$ (7.50$\times {10}^{-4}$)+	1.2242$\times {10}^{-1}$ (1.89$\times {10}^{-3}$)
	HV	1.0882$\times {10}^{-4}$ (2.18$\times {10}^{-4}$)−	5.4657$\times {10}^{-1}$ (1.17$\times {10}^{-3}$)+	5.5171$\times {10}^{-1}$ (1.55$\times {10}^{-3}$)+	5.3486$\times {10}^{-1}$ (1.72$\times {10}^{-3}$)+	1.3891$\times {10}^{-1}$ (2.78$\times {10}^{-1}$)−	5.6002$\times {10}^{-1}$ (9.06$\times {10}^{-4}$)+	5.1812$\times {10}^{-1}$ (3.13$\times {10}^{-3}$)
LIRCMOP14	IGD	1.2835$\times {10}^0$ (9.57$\times {10}^{-4}$)−	1.1296$\times {10}^{-1}$ (2.20$\times {10}^{-3}$)+	1.0104$\times {10}^{-1}$ (1.28$\times {10}^{-3}$)+	9.8191$\times {10}^{-2}$ (6.51$\times {10}^{-4}$)+	9.7651$\times {10}^{-1}$ (5.89$\times {10}^{-1}$)≈	9.5164$\times {10}^{-2}$ (5.54$\times {10}^{-4}$)+	1.1578$\times {10}^{-1}$ (2.34$\times {10}^{-3}$)
	HV	6.8210$\times {10}^{-4}$ (3.69$\times {10}^{-4}$)−	5.4635$\times {10}^{-1}$ (4.98$\times {10}^{-4}$)+	5.5153$\times {10}^{-1}$ (1.53$\times {10}^{-3}$)+	5.4960$\times {10}^{-1}$ (1.18$\times {10}^{-3}$)+	1.3948$\times {10}^{-1}$ (2.78$\times {10}^{-1}$)−	5.5519$\times {10}^{-1}$ (2.66$\times {10}^{-4}$)≈	5.3737$\times {10}^{-1}$ (1.60$\times {10}^{-3}$)
$+/-/\approx$	0/27/1	6/18/4	4/18/6	11/9/08	0/18/10	3/21/4

* Data with a dark background shows that the algorithm has reached the optimal value for the relevant problem index.

presents the experimental results of each algorithm. Across all test cases, DDCRO achieved 18 optimal results, URCMO obtained 6 optimal results, and CMODE-FTR secured 4 optimal results, while the remaining algorithms failed to obtain any optimal results. In contrast to DDCRO, CCMO’s auxiliary population focuses exclusively on searching for the UPF, which limits its effectiveness in scenarios where the UPF and CPF do not coincide. DDCRO’s dual-stage optimization strategy allows it to adapt to various constraint scenarios more effectively, particularly in problems with fragmented feasible regions like LIRCMOP1-4. Different from DDCRO, URCMO ensures that the algorithm converges to the CPF by learning the relationship between the UPF and CPF, and CAEAD ensures the distribution of the population on the CPF by adjusting the evolution and degradation strategies of its auxiliary population. However, DDCRO’s structured approach to deal with constraints through its molecular collision operator and weak complementarity mechanism provides a more balanced solution in the case of mixed CPF.

To provide a more intuitive comparison of the population characteristics among the algorithms, Figure 5 presents the population distribution of each algorithm on the LIR-CMOP8 problem. It can be observed that DDCRO is capable of covering all feasible regions and the CPF by combining global diversity search in the first phase with localized intensive search in the second phase. In NSGA-II, the population evidently fails to traverse large infeasible regions. While other algorithms can cross these regions, they show insufficient population diversity, which indicates their insufficient ability to explore CPF. CMODE-FTR can also cover CPF well by dynamically adjusting the preferences of the population between objective optimization and constraint satisfaction. However, some individuals still get trapped in local optima.

4.3. Comparisons on Real-World Problems

To validate the efficacy of the proposed DDCRO algorithm in addressing real-world challenges, this study conducted comparative experiments on real-world problems. Specifically, we evaluated DDCRO alongside contrast algorithms on eight representative sub-problems from the RWMOP test suite, which encompass practical issues spanning diverse domains within the RWMOP framework. Given that the true PF for these CMOPs remains unknown, we employed the HV metric to assess algorithm performance. The detailed experimental outcomes are presented in

Table 6. HV values obtained by seven algorithms on RWMOP test suite.

Problem		NSGAII	CAEAD	CCMO	URCMO	TSTI	CMODE-FTR	DDCRO
RWMOP5	HV	2.4048$\times {10}^{-1}$ (6.85$\times {10}^{-2}$)−	1.0000$\times {10}^0$ (0.00$\times {10}^0$)≈	3.6357$\times {10}^{-1}$ (4.36$\times {10}^{-2}$)−	2.6605$\times {10}^{-1}$ (1.87$\times {10}^{-2}$)−	3.8238$\times {10}^{-1}$ (1.57$\times {10}^{-2}$)−	9.5594$\times {10}^{-1}$ (6.75$\times {10}^{-2}$)+	5.1425$\times {10}^{-1}$ (4.41$\times {10}^{-2}$)
RWMOP7	HV	4.8535$\times {10}^{-1}$ (3.21$\times {10}^{-1}$)−	9.1229$\times {10}^{-1}$ (1.41$\times {10}^{-1}$)≈	4.7768$\times {10}^{-1}$ (1.44$\times {10}^{-3}$)−	4.8150$\times {10}^{-1}$ (3.68$\times {10}^{-4}$)−	4.7958$\times {10}^{-1}$ (1.11$\times {10}^{-3}$)−	7.1772$\times {10}^{-1}$ (3.22$\times {10}^{-1}$)≈	9.2924$\times {10}^{-1}$ (5.09$\times {10}^{-2}$)
RWMOP9	HV	3.3229$\times {10}^{-1}$ (1.41$\times {10}^{-1}$)−	4.6776$\times {10}^{-1}$ (1.94$\times {10}^{-1}$)−	5.4123$\times {10}^{-2}$ (8.79$\times {10}^{-4}$)−	5.5053$\times {10}^{-2}$ (1.09$\times {10}^{-3}$)−	5.4269$\times {10}^{-2}$ (9.68$\times {10}^{-4}$)−	1.7725$\times {10}^{-1}$ (3.24$\times {10}^{-3}$)−	9.2311$\times {10}^{-1}$ (8.25$\times {10}^{-2}$)
RWMOP10	HV	9.3570$\times {10}^{-1}$ (5.35$\times {10}^{-3}$)+	5.6658$\times {10}^{-1}$ (2.82$\times {10}^{-1}$)−	1.4404$\times {10}^{-1}$ (1.38$\times {10}^{-2}$)−	1.4175$\times {10}^{-1}$ (4.35$\times {10}^{-3}$)−	1.4478$\times {10}^{-1}$ (7.54$\times {10}^{-3}$)−	1.0000$\times {10}^0$ (0.00$\times {10}^0$)+	8.3282$\times {10}^{-1}$ (9.00$\times {10}^{-3}$)
RWMOP11	HV	5.8765$\times {10}^{-2}$ (8.48$\times {10}^{-3}$)−	2.6726$\times {10}^{-4}$ (3.72$\times {10}^{-4}$)−	7.1295$\times {10}^{-2}$ (2.56$\times {10}^{-3}$)−	1.7321$\times {10}^{-2}$ (4.14$\times {10}^{-3}$)−	4.1360$\times {10}^{-2}$ (1.84$\times {10}^{-2}$)−	1.0962$\times {10}^{-1}$ (3.06$\times {10}^{-1}$)−	8.9073$\times {10}^{-1}$ (0.00$\times {10}^0$)
RWMOP21	HV	8.2328$\times {10}^{-2}$ (2.16$\times {10}^{-3}$)+	9.3902$\times {10}^{-2}$ (4.13$\times {10}^{-2}$)+	2.9337$\times {10}^{-2}$ (1.35$\times {10}^{-5}$)+	2.9341$\times {10}^{-2}$ (2.10$\times {10}^{-5}$)+	3.0675$\times {10}^{-2}$ (9.40$\times {10}^{-5}$)+	3.7814$\times {10}^{-1}$ (1.98$\times {10}^{-1}$)+	2.6415$\times {10}^{-2}$ (2.68$\times {10}^{-3}$)
RWMOP25	HV	2.3374$\times {10}^{-1}$ (9.14$\times {10}^{-4}$)−	4.7956$\times {10}^{-1}$ (2.25$\times {10}^{-1}$)−	2.3656$\times {10}^{-1}$ (3.68$\times {10}^{-3}$)−	2.3689$\times {10}^{-1}$ (9.66$\times {10}^{-4}$)−	2.3765$\times {10}^{-1}$ (1.55$\times {10}^{-3}$)−	3.6364$\times {10}^{-1}$ (4.27$\times {10}^{-1}$)−	9.9735$\times {10}^{-1}$ (1.03$\times {10}^{-2}$)
RWMOP27	HV	2.8523$\times {10}^6$ (7.58$\times {10}^6$)≈	9.8338$\times {10}^{-1}$ (3.22$\times {10}^{-2}$)−	1.6764$\times {10}^6$ (7.50$\times {10}^6$)−	4.1701$\times {10}^5$ (1.48$\times {10}^6$)−	3.1610$\times {10}^6$ (8.43$\times {10}^6$)≈	3.5569$\times {10}^{-1}$ (4.11$\times {10}^{-1}$)−	3.7459$\times {10}^6$ (8.05$\times {10}^6$)
$+/-/\approx$	2/5/1	1/5/2	1/7/0	1/7/0	1/6/1	3/4/1

* Data with a dark background shows that the algorithm has reached the optimal value for the relevant problem index.

, which reveals that DDCRO attained the top performance in five out of the eight test scenarios. This is followed by CMODE-FTR, which secured the best results in two scenarios, and CAEAD, which excelled in one. The remaining algorithms did not achieve optimal performance in any of the tested scenarios. These empirical findings underscore the significant advantages of DDCRO over the comparison algorithms when tackling real-world problems. Consequently, the proposed DDCRO exhibits considerable practical significance and is well-suited for application in scientific and engineering contexts.

4.4. Effect Conflict Analysis of Molecular Collisions

When a molecule undergoes multiple on-wall ineffective collisions in the early stages, it gradually approaches a local optimum. However, if the kinetic energy (KE) of molecules is continuously reduced to meet the bi-molecular collision conditions, the current molecular structure will change significantly. Consequently, the well-optimized solution structure obtained through uni-molecule collision is disrupted, which results in the loss of previously acquired local optimal information. This weakens the algorithm’s local exploitation capability in the current region and affects the continuity of the search and the precision of the solution. To validate the aforementioned analysis, this experiment aims to uncover the conflict between the on-wall ineffective collision and bi-molecular reaction in CRO. We designed experiments such that molecules could only trigger either on-wall ineffective collision and synthesis reaction, or on-wall ineffective collision and inter-molecular ineffective collision during the evolutionary process. By observing the changes of molecular energy states, as depicted in Figure 6 and Figure 7, we assess the reasons for the impaired stability of the solution structure during the local search process.

As shown in Figure 6, the $PE$ and $KE$ of the molecule continue to decline until the eleventh collision, when both $PE$ and $KE$ suddenly surge. According to the reconstruction characteristics of molecular collisions, continuous on-wall ineffective collisions reduce the molecule’s $PE$ and $KE$, which enables the algorithm to continuously approach a better solution in the local region. However, the sustained decay of $KE$ triggers the conditions for a synthesis reaction. The fusion of molecules destroys the local optimal solution structure and wastes computational resources for searching. Figure 7 reveals that after the fourth molecular collision, there is a sudden increase in $KE$, followed by an abnormal rise in $PE$ during the fifth collision. Similar fluctuations reappear during the seventh and eighth collisions. It can be inferred that inter-molecular ineffective collisions endow molecules with higher $KE$, but they promote the generation of offspring molecules with significantly increased $PE$ in subsequent on-wall ineffective collisions. This causes the offspring to deviate from the current optimization trajectory, which weakens the coherence of the local search.

In summary, there exists a certain degree of search effect conflict between on-wall ineffective collisions and synthesis reactions, as well as between on-wall ineffective collisions and inter-molecular ineffective collisions during continuous execution. This may affect the search stability and the ability to maintain the solution structure of the CRO algorithm within local regions. Therefore, in the design of the CRO algorithm, it is necessary to reasonably coordinate the occurrence of various collision reactions to ensure the search efficiency and overall stability of the algorithm.

4.5. Ablation Study

In order to comprehensively evaluate the importance of each innovative component in DDCRO, we designed five variants and conducted ablation experiments on the MW test suite. Experiments include verifying the effectiveness of the dual-stage optimization mechanism, directional collision operators, and structure repair strategy. The five variants are DDCRO-noDS, DDCRO-stage1, DDCRO-stage2, DDGA, and DDCRO-noSRS. Specifically, DDCRO-noDS does not adopt a dual-stage optimization mechanism. DDCRO-stage1 only undergoes the first stage of evolution, and DDCRO-stage2 only undergoes the second stage of evolution. DDGA employs a dual-stage evolution but utilizes traditional GA operators for offspring generation in each stage, and DDCRO-noSRS abandons the structure repair strategy. It can be observed from Figure 8 that DDCRO-noDS, DDCRO-stage1, and DDCRO-stage2 exhibit relatively large IGD values across various problems. This is attributed to the fact that the effect conflict among collision operators cannot be effectively avoided without employing the dual-stage mechanism, and relying solely on either the first or second stage fails to achieve a balance between population convergence and diversity. The performance of DDGA is obviously worse than that of DDCRO, which shows that the offspring generation operator will also affect the performance of the algorithm in the dual-stage optimization process. The synergy of convergence and diversity collision operator is more effective than using the genetic operator alone to enhance the adaptability of the population in the evolution process. In contrast, DDCRO-noSRS and DDCRO achieve smaller IGD values on various problems. However, DDCRO-noSRS does not retain the structural information of the historical optimal solution obtained in the evolution process, which leads to its performance degradation compared with DDCRO. Through the aforementioned ablation experiments, we have vividly demonstrated the significance and efficacy of each innovative component within the DDCRO in preserving high-quality solutions.

5. Conclusions

This paper introduces a dual-stage and dual-population chemical reaction optimization (DDCRO) algorithm tailored to address CMOPs. The algorithm effectively balances feasibility and diversity by integrating the collision reaction mechanism of CRO with a dual-stage optimization strategy and a dual-population framework. Key innovations of DDCRO include refined molecular collision operators with a structure repair strategy, which preserve high-quality solutions during decomposition and synthesis reactions. Additionally, it features two directional search operators: convergence collision operators for enhancing local exploitation and diversity collision operators for boosting global exploration. The dual-stage optimization strategy divides population evolution into an initial phase focused on objective optimization to promote diversity, followed by a second phase emphasizing constraint satisfaction to expedite convergence to the CPF. The dual-population mechanism employs a main population and an auxiliary population to manage constrained and unconstrained problems separately, with a weak complementary mechanism facilitating information exchange between them to improve search efficiency and robustness. Extensive experiments on test suites demonstrate that DDCRO surpasses several state-of-the-art algorithms in terms of convergence speed, diversity preservation, and coverage of feasible solutions. First, DDCRO achieves optimal experimental results in 53% of test problems. Second, DDCRO’s staged optimization strategy effectively navigates through infeasible regions, leading to a more uniform distribution of solutions along the CPF. Lastly, by introducing the structural repair strategy, DDCRO ensures the maintenance of high-quality solutions throughout the decomposition and synthesis process.

However, the algorithm may experience a decline in performance when tackling high-dimensional problems or CMOPs characterized by highly nonlinear constraints. Future research will focus on extending DDCRO to address high-dimensional CMOPs and practical applications. To address high-dimensional problems, future research can explore the adoption of more efficient decomposition strategies or multi-objective dimension reduction techniques to maintain the efficiency and effectiveness of the algorithm. To integrate DDCRO with practical applications, tailored versions for specific application domains can be developed. By incorporating domain knowledge and constraints, the algorithm can better deal with the complexity and diversity of the real world. Additionally, it is necessary to further improve the dynamic threshold for stage switching to enhance the adaptability of the algorithm. In summary, DDCRO offers an effective solution for constrained multi-objective optimization, and its innovative framework provides valuable insights for future research.