ABSTRACT Balancing convergence, diversity and feasibility remains a fundamental challenge in constrained optimization. This paper proposes MOEA/D‐PS, which introduces the Pascoletti‐Serafini (PS) scalarization framework into constrained single‐objective optimization for the first time. The algorithm reformulates constrained single‐objective problems into a bi‐objective optimization by treating constraint violation as a secondary objective, then integrates two synergistic mechanisms to achieve a satisfactory balance of convergence, diversity and feasibility: (1) direction vectors sampled from a normal distribution to guide the population search around the elite solution while preserving exploration capability; (2) an adaptive truncation parameter that dynamically adjusts the search region based on the current elite solution. Unlike standard multi‐objective transformations that waste resources approximating the entire Pareto front, the PS‐guided mechanism geometrically restricts exploration to the trajectory toward the feasible optimum, achieving a unified balance among convergence, diversity and feasibility. Comprehensive experiments on 24 benchmark functions from the CEC 2006 test suite show that MOEA/D‐PS attains the best overall ranking among seven state‐of‐the‐art algorithms.

ABSTRACT Federated learning (FL) faces significant challenges under Non‐IID data, primarily due to misaligned local gradients that point in conflicting directions across clients. This inconsistency creates a fundamental trade‐off between global model accuracy and fairness, as clients with large‐magnitude gradients often dominate the aggregation process, marginalising minority participants. Existing methods typically fail to explicitly resolve these directional conflicts, leading to suboptimal convergence or exacerbated bias. To address this, we propose FedBEF, a theoretically grounded aggregation framework that jointly optimises convergence speed and fairness. Unlike heuristic weighting schemes, FedBEF derives optimal client weights by minimising an upper bound of the global loss, formulated as a constrained quadratic program. The resulting weights explicitly penalise gradient conflicts (e.g., by discouraging large angular deviations) while promoting a coherent descent direction that aligns with the true global gradient. Moreover, to improve robustness against noisy or outlier updates—particularly pronounced under partial client participation and extreme heterogeneity, we introduce a Similar Neighbour Gradient (SNG) mechanism combined with adaptive momentum. By clustering clients based on gradient cosine similarity and smoothing within neighbourhoods, SNG effectively suppresses erratic updates without requiring additional communication. More importantly, we prove that FedBEF converges to a stationary point under non‐convex settings. Extensive experiments on CIFAR‐10 and CIFAR‐100 demonstrate its superiority over state‐of‐the‐art baselines such as FedProx and FedALA. Notably, under severe heterogeneity (CIFAR‐10, Dirichlet ), FedBEF achieves up to a 7.13% higher average accuracy. It also significantly enhances fairness, as evidenced by a substantial reduction in the coefficient of variation (standard deviation over mean) of per‐client accuracies.

Visual prompt tuning (VPT), i.e., fine-tuning some lightweight prompt tokens, provides an efficient and effective approach for adapting pre-trained models to various downstream tasks. However, most prior art indiscriminately uses a fixed prompt distribution across different tasks, neglecting the importance of each block varying depending on the task. In this paper, we introduce adaptive distribution optimization (ADO) by tackling two key questions: (1) How to appropriately and formally define ADO, and (2) How to design an adaptive distribution strategy guided by this definition? Through empirical analysis, we first confirm that properly adjusting the distribution significantly improves VPT performance, and further uncover a key insight that a nested relationship exists between ADO and VPT. Based on these findings, we propose a new VPT framework, termed PRO-VPT (iterative Prompt RelOcation-based VPT), which adaptively adjusts the distribution built upon a nested optimization formulation. Specifically, we develop a prompt relocation strategy derived from this formulation, comprising two steps: pruning idle prompts from prompt-saturated blocks, followed by allocating these prompts to the most prompt-needed blocks. By iteratively performing prompt relocation and VPT, our proposal can adaptively learn the optimal prompt distribution in a nested optimization-based manner, thereby unlocking the full potential of VPT. Extensive experiments demonstrate that our proposal significantly outperforms advanced VPT methods, e.g., PRO-VPT surpasses VPT by 1.6 pp and 2.0 pp average accuracy, leading prompt-based methods to state-of-the-art performance on VTAB-1k and FGVC benchmarks. The code is available at https://github.com/ckshang/PRO-VPT.

Evolutionary multi-objective optimization (EMO) algorithms generate diverse Pareto optimal solutions to comprehensively approximate the Pareto front. However, an excessive number of diverse solutions pose significant challenges for decision makers who must select a single or few solutions for implementation. To address this decision-making burden, we propose a novel approach that promotes consistency among decision variables while accepting controlled performance trade-offs. Specifically, we develop VCM2M, an EMO algorithm that integrates a variable consistency strategy into the MOEA/D framework. It decomposes a multi-objective optimization problem (MOP) into a number of subproblems. In each subproblem, we incorporate the logarithmic sum regularization term of the decision variables into objective functions, achieving consistency during both the optimization and selection phases. An ϵ-insensitive loss function is utilized to normalize the loss of population diversity in the objective space. Experimental evaluation on specially constructed test problems demonstrates that VCM2M successfully generates solution sets organized into distinct clusters, where solutions within each cluster exhibit significantly higher variable consistency compared to those produced by MOEA/D-M2M, AR-MOEA, and NSGA-III.

ABSTRACT Clustering analysis is one of the most important data mining and knowledge discovery tools in real applications. Since the widespread presence of missing values hampers clustering performance, missing values imputation becomes necessary for data pre‐processing. However, for the common datasets composed of both numerical and categorical attributes (also known as mixed‐attribute datasets), most existing imputation methods suffer from the following three limitations: (1) Only feasible for a certain type of attribute; (2) Encounter difficulties in considering the interdependence between different types of attributes; (3) Short in exploiting the information provided by the incomplete mix‐valued objects. As a result, the original data distribution can be ill‐restored, misleading the downstream clustering tasks. This paper therefore proposes a clustering‐imputation co‐learning method for incomplete mixed‐attribute datasets to address these issues. This method integrates imputation and clustering into one learning process, emphasising the interrelationships between mixed attributes during the imputation process and exploiting the information of incomplete objectsduring clustering. It turns out that appropriate recovery of the dataset and accurate clustering can be better achieved through a cross‐coupling manner. Experiments on various datasets validate the promising efficacy of the proposed method.

Pareto Set Learning (PSL) is an emerging research area in multi-objective optimization, focusing on training neural networks to learn the mapping from preference vectors to Pareto optimal solutions. However, existing PSL methods are limited to addressing a single Multi-objective Optimization Problem (MOP) at a time. When faced with multiple MOPs, this limitation results in significant inefficiencies and hinders the ability to exploit potential synergies across varying MOPs. In this paper, we propose a Collaborative Pareto Set Learning (CoPSL) framework, which learns the Pareto sets of multiple MOPs simultaneously in a collaborative manner. CoPSL particularly employs an architecture consisting of shared and MOP-specific layers. The shared layers are designed to capture commonalities among MOPs collaboratively, while the MOP-specific layers tailor these general insights to generate solution sets for individual MOPs. This collaborative approach enables CoPSL to efficiently learn the Pareto sets of multiple MOPs in a single execution while leveraging the potential relationships among various MOPs. To further understand these relationships, we experimentally demonstrate that shareable representations exist among MOPs. Leveraging these shared representations effectively improves the capability to approximate Pareto sets. Extensive experiments underscore the superior efficiency and robustness of CoPSL in approximating Pareto sets compared to state-of-the-art approaches on a variety of synthetic and real-world MOPs. Code is available at https://github.com/ckshang/CoPSL.

Multiobjective multitask optimization (MMO) attempts to solve several problems simultaneously. This is commonly done by identifying useful knowledge to transfer between tasks, thereby producing optimal solutions more quickly. In this study, an MMO algorithm using transfer rank and a KNN model is proposed to achieve this goal. The definition of transfer rank is first introduced for quantifying the priority of transfer solutions, to improve the probability of a positive result. The solution with the higher rank was assumed to be the most suitable for transfer, as solutions were sorted in descending order based on transfer rank. Priority was given to previous and positive-transfer solutions and those with the same transfer rank were distinguished using a KNN model classifier. The effectiveness of the proposed algorithm was verified by studying benchmark MMO problems. The experimental results showed the proposed algorithm was more effective than other conventional MMO techniques.

As many real data sets (e.g., social, financial, and medical data sets) are successively generated in evolution with the ever-changing environment, classification for data stream with concept drift attracts increasing attention in the fields of machine learning and data mining. However, to the best of our knowledge, existing works mainly consider the concept drift issue while ignoring another common characteristic of real data, i.e., existence of awkward heterogeneity caused by mixture of numerical and categorical attributes. It is worth noting that tackling both the concept drift and heterogeneity problems together is exponentially more challenging than dealing with only one of them. This paper, therefore, proposes an ensemble learning approach for the classification of numerical-and-categorical-attribute data (also called mixed data hereinafter) under concept drift. We first design a unified metric to appropriately address the heterogeneity of numerical and categorical attributes. Then a base classifier that can appropriately fuse the information provided by the heterogeneous attributes is formed accordingly. Furthermore, to make the classification adapt to the complex concept drifts demonstrated on the heterogeneous attributes, two types of base classifier ensembles are dynamically learned on the fly. Experimental results on various real mixed data sets with concept drifts demonstrate the efficacy of the proposed method.

This article presents a rough-to-fine evolutionary multiobjective optimization algorithm based on the decomposition for solving problems in which the solutions are initially far from the Pareto-optimal set. Subsequently, a tree is constructed by a modified k -means algorithm on N uniform weight vectors, and each node of the tree contains a weight vector. Each node is associated with a subproblem with the help of its weight vector. Consequently, a subproblem tree can be established. It is easy to find that the descendant subproblems are refinements of their ancestor subproblems. The proposed algorithm approaches the Pareto front (PF) by solving a few subproblems in the first few levels to obtain a rough PF and gradually refining the PF by involving the subproblems level-by-level. This strategy is highly favorable for solving problems in which the solutions are initially far from the Pareto set. Moreover, the proposed algorithm has lower time complexity. Theoretical analysis shows the complexity of dealing with a new candidate solution is O(M logN) , where M is the number of objectives. Empirical studies demonstrate the efficacy of the proposed algorithm.

Numerous surrogate-assisted expensive multi-objective optimization algorithms were proposed to deal with expensive multi-objective optimization problems in the past few years. The accuracy of the surrogate models degrades as the number of decision variables increases. In this paper, we propose a surrogate-assisted expensive multi-objective optimization algorithm based on decision space compression. Several surrogate models are built in the lower dimensional compressed space. The promising points are generated and selected in the lower compressed decision space and decoded to the original decision space for evaluation. Experimental studies show that the proposed algorithm achieves a good performance in handling expensive multi-objective optimization problems with high-dimensional decision space.

Multiobjective multitasking optimization (MTO), which is an emerging research topic in the field of evolutionary computation, was recently proposed. MTO aims to solve related multiobjective optimization problems at the same time via evolutionary algorithms. The key to MTO is the knowledge transfer based on sharing solutions across tasks. Notably, positive knowledge transfer has been shown to facilitate superior performance characteristics. However, how to find more valuable transferred solutions for the positive transfer has been scarcely explored. Keeping this in mind, we propose a new algorithm to solve MTO problems. In this article, if a transferred solution is nondominated in its target task, the transfer is positive transfer. Furthermore, neighbors of this positive-transfer solution will be selected as the transferred solutions in the next generation, since they are more likely to achieve the positive transfer. Numerical studies have been conducted on benchmark problems of MTO to verify the effectiveness of the proposed approach. Experimental results indicate that our proposed framework achieves competitive results compared with the state-of-the-art MTO frameworks.

This article investigates the properties of ratio and difference-based indicators under the Minkovsky distance and demonstrates that a ratio-based indicator with infinite norm is the best for solution evaluation among these indicators. Accordingly, a promising-region-based evolutionary many-objective algorithm with the ratio-based indicator is proposed. In our proposed algorithm, a promising region is identified in the objective space using the ratio-based indicator with infinite norm. Since the individuals outside the promising region are of poor quality, we can discard these solutions from the current population. To ensure the diversity of population, a strategy based on the parallel distance is introduced to select individuals in the promising region. In this strategy, all individuals in the promising region are projected vertically onto the normal plane so that crowded distances between them can be calculated. Afterward, two solutions with a smaller distance are selected from the candidate solutions each time, and the solution with the smaller indicator fitness value is removed from the current population. Empirical studies on various benchmark problems with 3-20 objectives show that the proposed algorithm performs competitively on all test problems. Compared with a number of other state-of-the-art evolutionary algorithms, the proposed algorithm is more robust on these problems with various Pareto fronts.

Prediction-based evolutionary multi-objective optimization algorithm is one of the most popular optimization algorithms for solving dynamic multi-objective optimization problem. It uses time-series models to predict the future Pareto set based on the past solutions. However, the dimension of the decision variables may be too high to predict. Moreover, a relatively small variance in decision variables may lead to a large difference in the objective space. The optimized Pareto front (PF) may be far from the desired output. To solve these problems, this paper proposes a new co-operative prediction method, which predicts not only the Pareto solution (PS), but also a hyper-plane as an approximation of the prediction of the PF in the objective space. The hyper-plane is used to guide the search process and accelerate the convergence. We compare the proposed algorithm with three existing dynamic optimization algorithms. Experimental results show the effectiveness of the proposed algorithm.

This paper addresses a class of optimization problems in which either part of the objective function is differentiable while the rest is nondifferentiable or the objective function is differentiable in only part of the domain. Accordingly, we propose a dual-decomposition-based approach that includes both objective decomposition and domain decomposition. In the former, the original objective function is decomposed into several relatively simple subobjectives to isolate the nondifferentiable part of the objective function, and the problem is consequently formulated as a multiobjective optimization problem (MOP). In the latter decomposition, we decompose the domain into two subdomains, that is, the differentiable and nondifferentiable domains, to isolate the nondifferentiable domain of the nondifferentiable subobjective. Subsequently, the problem can be optimized with different schemes in the different subdomains. We propose a population-based optimization algorithm, called the simulated water-stream algorithm (SWA), for solving this MOP. The SWA is inspired by the natural phenomenon of water streams moving toward a basin, which is analogous to the process of searching for the minimal solutions of an optimization problem. The proposed SWA combines the deterministic search and heuristic search in a single framework. Experiments show that the SWA yields promising results compared with its existing counterparts.

Multiobjective multitasking optimization (MTO) is an emerging research topic in the field of evolutionary computation. In contrast to multiobjective optimization, MTO solves multiple optimization tasks simultaneously. MTO aims to improve the overall performance of multiple tasks through knowledge transfer among tasks. Recently, MTO has attracted the attention of many researchers, and several algorithms have been proposed in the literature. However, one of the crucial issues, finding useful knowledge, has been rarely studied. Keeping this in mind, this article proposes an MTO algorithm based on incremental learning (EMTIL). Specifically, the transferred solutions (the form of knowledge) will be selected by incremental classifiers, which are capable of finding valuable solutions for knowledge transfer. The training data are generated by the knowledge transfer at each generation. Furthermore, the search space of the tasks will be explored by the proposed mapping (among tasks) approach, which helps these tasks to escape from their local Pareto Fronts. Empirical studies have been conducted on 15 MTO problems to assess the effectiveness of EMTIL. The experimental results demonstrate that EMTIL works more effectively for MTO compared to the existing algorithms.

Based on the ideas of minimizing the loss of convergence and diversity of the candidate solution set, this paper proposes a cost value based evolutionary many-objective algorithm with neighbor selection strategy. In this work, the cost value of each solution is the mutual evaluation from other ones in current population. By this way, the proposed algorithm, named MEMO, can easily recognize the dominated and the nondominated solutions and assess the contribution of convergence and diversity of each solution among the candidate solution set. To further enhance the performance of proposed algorithm, a neighbor selection strategy is also suggested in this paper. Simulation experiments on MaF series indicate that the proposed MEMO is superior to IBEA, MOEA/D, NSGA-III and RVEA in terms of effectiveness and robustness.

Many-objective optimization problems (MaOPs), in which the number of objectives is greater than three, are undoubtedly more challenging compared with the bi- and tri-objective optimization problems. Currently, the decomposition-based evolutionary algorithms have shown promising performance in dealing with MaOPs. Nevertheless, these algorithms need to design the weight vectors, which has significant effects on the performance of the algorithms. In particular, when the Pareto front of problems is incomplete, these algorithms cannot obtain a set of uniformly distribution solutions by using the conventional weight design methods. In the literature, it is well-known that the self-organizing map (SOM) can preserve the topological properties of the input data by using the neighborhood function, and its display is more uniform than the probability density of the input data. This phenomenon is advantageous to generate a set of uniformly distributed weight vectors based on the distribution of the individuals. Therefore, we will propose a novel weight design method based on SOM, which can be integrated with most of the decomposition-based algorithms for solving MaOPs. In this paper, we choose the existing state-of-the-art decomposition-based algorithms as examples for such integration. This integrated algorithms are then compared with some state-of-the-art algorithms on eleven redundancy problems and eight nonredundancy problems, respectively. The experimental results show the effectiveness of the proposed approach.

In this paper, we propose a simple but effective objective reduction algorithm (ORA) for many-objective optimization problems (MaOPs). It uses a hyperplane involving sparse non-negative coefficients to roughly approximate the conflicting structure of the Pareto front in the objective space. Then the objectives with non-zero coefficients are considered as essential objectives. In order to verify the performance of proposed algorithm, we compare the proposed algorithm with two correlation-based ORA, i.e., L-PCA and NL-MVU-PCA and a dominance structure-based ORA, i.e., PCSEA on the benchmark problem DTLZ5(I, M). The experimental results show the effectiveness of the proposed algorithm.

This paper proposes two constraint-handling techniques based on multiobjective optimization with biased dynamic weights for constrained optimization problems (COPs). Transforming a COP into an unconstrained biobjective optimization, two popular strategies based on decomposition, i.e. Tchebycheff approach (TEA) and weighted sum approach (WSA) are used in this paper respectively. In order to keep a good balance between convergence and diversity of the population, this paper uses the weights, which are designed with bias and change dynamically as the generation increases, to select different individuals with smaller objective values and lower degree of constraint violations. Furthermore, 13 benchmark test functions are used to investigate the effectiveness of TEA and WSA. Experimental results demonstrate that TEA not only works better than WSA, but also is superior to the compared algorithms, i.e. MDPE, GDE and SR in terms of reliability and stabilization of converging to a global solution.

In this paper, we propose a decomposition-based evolutionary algorithm with boundary search and archive for constrained multi-objective optimization problems (CMOPs), named CM2M. It decomposes a CMOP into a number of optimization subproblems and optimizes them simultaneously. Moreover, a novel constraint handling scheme based on the boundary search and archive is proposed. Each subproblem has one archive, including a subpopulation and a temporary register. Those individuals with better objective values and lower constraint violations are recorded in the subpopulation, while the temporary register consists of those individuals ever found before. To improve the efficiency of the algorithm, the boundary search method is designed. This method makes the feasible individuals with a higher probability to perform genetic operator with the infeasible individuals. Especially, when the constraints are active at the Pareto solutions, it can play its leading role. Compared with two algorithms, i.e. CMOEA/D-DE-CDP and Gary’s algorithm, on 18 CMOPs, the results show the effectiveness of the proposed constraint handling scheme.

This paper presents a niching-based evolutionary algorithm for optimizing multi-modal optimization function. Provided that the potential optima are characterized by a relatively smaller objective value than their neighbors and by a relatively large distance from points with smaller objective values, we identify potential optima from individuals. Using them as seeds, a population is decomposed into a number of subpopulations without introducing new parameters. Moreover, we present an adaptive allocating strategy of assigning different computational resources to different subpopulations upon the fact that discovering different optima may have different computational difficulty. The proposed method is compared with three state-of-the-art multi-modal optimization approaches on a benchmark function set. The extensive experimental results demonstrate its efficacy.

For many-objective optimization problems (MaOPs), in which the number of objectives is greater than three, the performance of most existing evolutionary multi-objective optimization algorithms generally deteriorates over the number of objectives. As some MaOPs may have redundant or correlated objectives, it is desirable to reduce the number of the objectives in such circumstances. However, the Pareto solution of the reduced MaOP obtained by most of the existing objective reduction methods, based on objective selection, may not be the Pareto solution of the original MaOP. In this paper, we propose an objective extraction method (OEM) for MaOPs. It formulates the reduced objective as a linear combination of the original objectives to maximize the conflict between the reduced objectives. Subsequently, the Pareto solution of the reduced MaOP obtained by the proposed algorithm is that of the original MaOP, and the proposed algorithm can thus preserve the dominance structure as much as possible. Moreover, we propose a novel framework that features both simple and complicated Pareto set shapes for many-objective test problems with an arbitrary number of essential objectives. Within this framework, we can control the importance of essential objectives. As there is no direct performance metric for the objective reduction algorithms on the benchmarks, we present a new metric that features simplicity and usability for the objective reduction algorithms. We compare the proposed OEM with three objective reduction methods, i.e., REDGA, L-PCA, and NL-MVU-PCA, on the proposed test problems and benchmark DTLZ5 with different numbers of objectives and essential objectives. Our numerical studies show the effectiveness and robustness of the proposed approach.

This paper presents a non-parameter method to identify the peaks of the multi-modal optimization problems provided that the peaks are characterized by a smaller objective values than their neighbors and by a relatively large distance from points with smaller objective value. Using the identified peaks as the seeds, we decompose the population into some subpopulations and dynamically allocate the computational effort to different subpopulations. We evaluate the proposed approach on the CEC2015 single objective multi-niche optimization problems. The promising experimental results show its efficacy.

For many-objective optimization problems, i.e. the number of objectives is greater than three, the performance of most of the existing Evolutionary Multi-objective Optimization algorithms will deteriorate to a certain degree. It is therefore desirable to reduce many objectives to fewer essential objectives, if applicable. Currently, most of the existing objective reduction methods are based on objective selection, whose computational process is, however, laborious. In this paper, we will propose an online objective reduction method based on objective extraction for the many-objective optimization problems. It formulates the essential objective as a linear combination of the original objectives with the combination weights determined based on the correlations of each pair of the essential objectives. Subsequently, we will integrate it into NSGA-II. Numerical studies have show the efficacy of the proposed approach.

As the usage of fingerprint systems is rolled out on a large scale, scenarios have cross-device matching to allow information exchange and provide compatibility to the existing systems. A score-level calibration for device interoperability will require normalizing scores obtained from different devices so that they can be matched meaningfully and effectively. Conventional methods either assume a homogeneous distribution or model score distribution based on assumptions that may not be valid. In this paper, we circumvent the problem by leveraging correlations among the scores and propose a novel method for biometric score normalization. Our experiments show the promising results.

This letter suggests an approach for decomposing a multiobjective optimization problem (MOP) into a set of simple multiobjective optimization subproblems. Using this approach, it proposes MOEA/D-M2M, a new version of multiobjective optimization evolutionary algorithm-based decomposition. This proposed algorithm solves these subproblems in a collaborative way. Each subproblem has its own population and receives computational effort at each generation. In such a way, population diversity can be maintained, which is critical for solving some MOPs. Experimental studies have been conducted to compare MOEA/D-M2M with classic MOEA/D and NSGA-II. This letter argues that population diversity is more important than convergence in multiobjective evolutionary algorithms for dealing with some MOPs. It also explains why MOEA/D-M2M performs better.

This paper addresses the complex optimization problem, of which the objective function consists of two parts: One part is differentiable and the other part is non-differentiable. Accordingly, we decompose the original objective function into several relatively simple sub-objective ones, which subsequently formulate as a multiobjective optimization problem (MOP). To solve this MOP, we propose a simulated water-stream algorithm (SWA) inspired by the natural phenomenon of water streams. The water streams with a hybrid process of downstream and penetration towards the basin is analogous to the process of finding the minimum solution in an optimization problem. The SWA featuring a combination of deterministic search and heuristic search generally converges much faster than the existing counterparts with a considerable accuracy enhancement. Experimental results show the efficacy of the proposed algorithm.

A novel multiobjective DE algorithm using the subregion and external set strategy (MOEA/S-DE) is proposed in this paper, in which the objective space is divided into some subregions and then independently optimize each subregion. An external set is introduced for each subregion to save some individuals ever found in this subregion. An alternative of mutation operators based the idea of direct simplex method of mathematical programming are proposed: local and global mutation operator. The local mutation operator is applied to improve the local search performance of the algorithm and the global mutation operator to explore a wider area. Additionally, a reusing strategy of difference vector also is proposed. It reuses the difference vector of the better individuals according to a given probability. Compared with traditional DE, the crossover operator also is improved. In order to demonstrate the performance of the proposed algorithm, it is compared with the MOEA/D-DE and the hybrid-NSGA-II-DE. The result indicates that the proposed algorithm is efficient.

By dividing the objective space into several small regions, this paper proposes an improved NSGA-II algorithm, which updates the population in each sub-region by using non-dominated sorting and crowded distance selection operator (NSGA-II). Since performing the evolutionary operator is independent in each sub-region and the number of the individuals in a sub-region is far less than the size of the population, the computational complexity at each generation is lower than NSGA-II. The computational complexity of each generation in the proposed algorithm is O(mN3/2), where m is the number of the objective and N is its population size. For enhancing the capability of proposed algorithm, The algorithm exchanges the information between sub-regions through re-dividing their offsprings and the evolutionary operators between individuals are operated in the same sub-region. Such evolutionary operators can largely play a role of exploring the good individuals in this area and improve the local search capabilities of the algorithm. A specific selection in this paper surmounts the intrinsic shortcoming of the sub region decomposition technique, which there may be no Pareto optimal solutions in some sub-region. Numerical results show that the proposed algorithm has a good performance.