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When solving constrained multiobjective optimization problems (CMOPs), the most commonly used way of measuring constraint violation is to calculate the sum of all constraint violations of a solution as its distance to feasibility. However, this kind of constraint violation measure may not reflect the distance of an infeasible solution from feasibility for some problems, for example, when an infeasible solution closer to a feasible region does not have a smaller constraint violation than the one farther away from a feasible region. Unfortunately, no set of artificial benchmark problems focusing on this area exists. To remedy this issue, a set of CMOPs with deceptive constraints is introduced in this article. It is the first attempt to consider CMOPs with deceptive constraints (DCMOPs). Based on our previous work, which designed a set of directed weight vectors to solve CMOPs, this article proposes a cooperative framework with an improved version of directed weight vectors to solve DCMOPs. Specifically, the cooperative framework consists of two switchable phases. The first phase uses two subpopulations-one to explore feasible regions and the other to explore the entire space. The two subpopulations provide useful information about the optimal direction of objective improvement to each other. The second phase aims mainly at finding Pareto-optimal solutions. Then an infeasibility utilization strategy is used to improve the objective function values. The two phases are switchable based on the information found to date at any time in the evolutionary process. The experimental results show that this method significantly outperforms the algorithms with which it is compared on most of the DCMOPs, in terms of reliability and stability in finding a set of well-distributed optimal solutions.

In this paper, we generalize the usual multiplicative ∊ -indicator I ∊ and additive ∊ -indicator I ∊ + to the forms with L p -norm, i.e. I ∊ p -indicator and I ∊ + p -indicator, and investigate their properties under different p. From the analysis, we find that both I ∊ p -indicator and I ∊ + p -indicator have a good performance on measuring the population convergence and diversity, especially when p is set to be infinity. In order to show their differences more intuitively, we implement a series of comparative experiments by the proposed I ∊ p -indicator and I ∊ + p -indicator based evolutionary algorithm. The experimental results confirm that all these ∊ -indicators have good performances, particulary as p is set to be infinity. We also compare the proposed indicator-based evolutionary algorithm with five existing algorithms. The experimental results demonstrate that our algorithm is effective. In addition, considering that the usual IGD metric is not accurate in measuring the convergence of a population, we adopt the measure based on I ∊ + p -indicator and propose a more effective performance metric, which is denoted as IGD ∊ + .

Transfer learning methods exploit similarities between different datasets to improve the performance of the target task by transferring knowledge from source tasks to the target task. “What to transfer” is a main research issue in transfer learning. The existing transfer learning method generally needs to acquire the shared parameters by integrating human knowledge. However, in many real applications, an understanding of which parameters can be shared is unknown beforehand. Transfer learning model is essentially a special multi-objective optimization problem. Consequently, this paper proposes a novel auto-sharing parameter technique for transfer learning based on multi-objective optimization and solves the optimization problem by using a multi-swarm particle swarm optimizer. Each task objective is simultaneously optimized by a sub-swarm. The current best particle from the sub-swarm of the target task is used to guide the search of particles of the source tasks and vice versa. The target task and source task are jointly solved by sharing the information of the best particle, which works as an inductive bias. Experiments are carried out to evaluate the proposed algorithm on several synthetic data sets and two real-world data sets of a school data set and a landmine data set, which show that the proposed algorithm is effective.

An objective normalization strategy is essential in any evolutionary multiobjective or many-objective optimization (EMO or EMaO) algorithm, due to the distance calculations between objective vectors required to compute diversity and convergence of population members. For the decomposition-based EMO/EMaO algorithms involving the Penalty Boundary Intersection (PBI) metric, normalization is an important matter due to the computation of two distance metrics. In this article, we make a theoretical analysis of the effect of instabilities in the normalization process on the performance of PBI-based MOEA/D and a proposed PBI-based NSGA-III procedure. Although the effect is well recognized in the literature, few theoretical studies have been done so far to understand its true nature and the choice of a suitable penalty parameter value for an arbitrary problem. The developed theoretical results have been corroborated with extensive experimental results on three to 15-objective convex and non-convex instances of DTLZ and WFG problems. The article, makes important theoretical conclusions on PBI-based decomposition algorithms derived from the study.

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.

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.

Hypervolume indicator based evolutionary algorithms have been reported to be very promising in many-objective optimization, but the high computational complexity of hypervolume calculation in high dimensions restrains its further applications and developments. In this paper, we develop a fast hypervolume approximation method with both improved speed and accuracy than the previous approximation methods via a new segmentation strategy. The proposed approach consists of two crucial process: segmentation and approximation. The segmentation process recursively finds areas easy to be measured and quantified from the original geometric figure as many as possible, and then divides the measurement of the rest areas into several subproblems. In the approximation process, an improved Monte Carlo simulation is developed to estimate these subproblems. Those two processes are mutually complementary to simultaneously improve the accuracy and the speed of hypervolume approximation. To validate its effectiveness, experimental studies on four widely-used instances are conducted and the simulation results show that the proposed method is ten times faster than other comparison algorithms with a same measurement error. Furthermore, we integrate an incremental version of this method into the framework of SMS-EMOA, and the performance of the integrated algorithm is also very competitive among the experimental algorithms.

Decomposition-based evolutionary algorithms have shown great potential in many-objective optimization. However, the lack of theoretical studies on decomposition methods has hindered their further development and application. In this paper, we first theoretically prove that weight sum, Tchebycheff, and penalty boundary intersection decomposition methods are essentially interconnected. Inspired by this, we further show that highly customized dominance relationship can be derived from decomposition for any given decomposition vector. A new evolutionary algorithm is then proposed by applying the customized dominance relationship with adaptive strategy to each subpopulation of multiobjective to multiobjective framework. Experiments are conducted to compare the proposed algorithm with five state-of-the-art decomposition-based evolutionary algorithms on a set of well-known scaled many-objective test problems with 5 to 15 objectives. Simulation results have shown that the proposed algorithm can make better use of the decomposition vectors to achieve better performance. Further investigations on unscaled many-objective test problems verify the robust and generality of the proposed algorithm.

Decomposition-based evolutionary multiobjective algorithms achieve good performance for solving the problems with regular Pareto fronts. Nevertheless, the shape of the Pareto front greatly influences the performance of the algorithms. Thus, we propose a new adaptive parallel region decomposition strategy. Different from the traditional decomposition-based methods, the proposed algorithm decomposes a multiobjective optimization problem into a number of subproblems by different ideal points, but not by different weight vectors. We compare the proposed algorithm with four state-of-the-art algorithms on seven test problems with irregular Pareto fronts. Experimental results show that the proposed algorithm has superior robustness on the optimization problems with irregular Pareto fronts.

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.

Energy efficiency of the wireless networks has drawn more and more attentions due to the requirement of the green communication. The base station sleeping strategy and resource allocation can effectively improve the energy efficiency for the wireless networks. Meanwhile, renewable energy is important to decrease the carbon emission. In this paper, the authors propose a joint BS sleeping strategy, resource allocation and renewable energy procurement scheme to maximize the profit of the network operators and minimize the carbon emission. Then, a joint optimization problem is formulated, which is a mixed integer programming problem. To solve it, they adopt the bi-velocity discrete particle swarm optimization (BVDPSO) algorithm to optimize the BS sleeping strategy. When the BS sleeping strategy is fixed, the authors propose an optimal algorithm based on Lagrange dual domain method to optimize the power allocation, subcarrier assignment and energy procurement. Numerical results illustrate the effectiveness of their proposed scheme and algorithm.

An effective allocation of search effort is important in multiobjective optimization, particularly in many-objective optimization problems (MaOPs). This paper presents a new adaptive search effort allocation strategy for multiobjective evolutionary algorithm based on decomposition MOEA/D-M2M, a recent MOEA/D algorithm for challenging MaOPs. This proposed method adaptively adjusts the subregions of its subproblems by detecting the importance of different objectives in an adaptive manner. More specifically, it periodically resets the subregion setting based on the distribution of the current solutions in the objective space such that the search effort is not wasted on unpromising regions. The basic idea is that the current population can be regarded as an approximation to the Pareto front (PF) and thus one can implicitly estimate the shape of the PF and such estimation can be used for adjusting the search focus. The performance of proposed algorithm has been verified by comparing it with eight representative and competitive algorithms on a set of degenerated MaOPs with disconnected and connected PFs. Performances of the proposed algorithm on a number of nondegenerated test instances with connected and disconnected PFs are also studied.

When solving constrained multi-objective optimization problems (CMOPs), keeping infeasible individuals with good objective values and small constraint violations in the population can improve the performance of the algorithms, since they provide the information about the optimal direction towards Pareto front. By taking the constraint violation as an objective, we propose a novel constraint-handling technique based on directed weights to deal with CMOPs. This paper adopts two types of weights, i.e. feasible and infeasible weights distributing on feasible and infeasible regions respectively, to guide the search to the promising region. To utilize the useful information contained in infeasible individuals, this paper uses infeasible weights to maintain a number of well-diversified infeasible individuals. Meanwhile, they are dynamically changed along with the evolution to prefer infeasible individuals with better objective values and smaller constraint violations. Furthermore, 18 test instances and 2 engineering design problems are used to evaluate the effectiveness of the proposed algorithm. Several numerical experiments indicate that the proposed algorithm outperforms four compared algorithms in terms of finding a set of well-distributed non-domination solutions.

Bi-objective constraint-handling technique may be one of the most promising constraint techniques for constrained optimization problems. It regards the constraints as an extra objective and using Pareto ranking as selection operator. These algorithms achieve a good convergence by utilizing potential infeasible individuals, but not be good at maintaining the diversity of the population. It is significant to balance the diversity of the population and the convergence of the algorithm. This paper proposes a novel constraint-handling technique based on biased dynamic weights for constrained evolutionary algorithm. The biased weights are used to select different individuals with low objective values and low degree of constraint violations. Furthermore, along with the evolution, more emphasis is placed on the individuals with lower objective values and lower degree of constraint violations by adjusting the biased weights dynamically, which forces the search to a promising feasible region. Thus, the proposed algorithm can keep a good balance between the convergence and the diversity of the population. Moreover, we compared the proposed algorithm with other state-of-the-art algorithms on 42 benchmark problems. The experimental results showed the reliability and stabilization of the proposed algorithm.

When planning the Tracking Areas (TAs) for a Long Term Evolution (LTE) network, the main concern of mobile operators is to achieve the minimization of both location update cost and paging cost. This paper proposes a new green field TA planning model using multi-objective optimization with constraints, aiming at finding a better trade-off between the two conflicting objectives. This new model integrates the network geographical information, therefore making it more realistic. Considering the impact of constraints, we design an evolutionary multi-objective algorithm based on a population decomposition strategy for the proposed model. Information about infeasible solutions can be fully utilized by population decomposition and thus the algorithmic efficiency can be greatly improved. A new coding scheme inspired by the famous four-color theorem is specially designed for this multi-objective TA planning model. Computer simulations are conducted and the quality of the new model is confirmed by comparing the results of the multi-objective model with those of a single-objective model. The essential role of the population decomposition strategy has also been identified by comparing the proposed algorithm with the Multi-objective Evolutionary Algorithm based on Decomposition (MOEA/D).

Energy efficiency of the wireless networks has drawn more and more attentions due to the requirement of the green communication. The base station sleeping strategy and resource allocation can effectively improve the energy efficiency for the wireless networks. Meanwhile, renewable energy is important to decrease the carbon emission. In this paper, the authors propose a joint BS sleeping strategy, resource allocation and renewable energy procurement scheme to maximize the profit of the network operators and minimize the carbon emission. Then, a joint optimization problem is formulated, which is a mixed integer programming problem. To solve it, they adopt the bi-velocity discrete particle swarm optimization (BVDPSO) algorithm to optimize the BS sleeping strategy. When the BS sleeping strategy is fixed, the authors propose an optimal algorithm based on Lagrange dual domain method to optimize the power allocation, subcarrier assignment and energy procurement. Numerical results illustrate the effectiveness of their proposed scheme and algorithm.

Unconstrained optimization problems, arise in many practical applications. Especially, significant improvement in deep learning training came from the Quasi-Newton methods. They exploit the idea of building up curvature information as the iterations of the training method are progressing. BFGS update in Quasi-Newton methods is the most commonly used update rule for training deep neural networks. The accuracy of computed search direction depends largely on how sensitive the Hessian approximation matrix Bk+1 is to small changes. The larger distribution of eigenvalues of the matrix will cause more sensation. In this paper, we propose a contracting BFGS update (C-BFGS), in order to contract the interval of distribution. The new update retains the Hessian approximation matrix positive definiteness, so that it makes sure the search direction down. For a quadratic positive definite function, the search directions generated by C-BFGS under the exact line search are Gconjugate, and the Quasi-Newton method with C-BFGS update satisfies quadratic termination property.

When using constrained multi-objective evolutionary algorithms to solve problems, most of them prefer infeasible solutions closer to feasible regions since they have smaller constraint violations. However, they may not work well on some problems which infeasible solutions closer to feasible regions have larger constraint violations than the ones farther away from the feasible regions. This kind of problems has not been explored so far. To help close this research gap, this paper first illustrates it by designing two new constrained multi-objective optimization problems (CMOPs) with deceptive infeasible regions. That is, in the deceptive infeasible regions, solutions farther away from a feasible region has a smaller constraint violation than the ones closer to the feasible region. We have proposed a constraint-handling technique based a set of directed weights (M2M-DW) to solve CMOPs on our previous work. However, it can not be able to solve the CMOPs with deceptive infeasible regions, since the directed weights prefer infeasible solutions with smaller constraint violations, which guides the search to the less promising regions. We improve it to solve the two proposed problems. Another set of benchmark problems, i.e., CTPs, is also used to verify the performance of the compared algorithms. Experimental results show that the proposed algorithm significantly outperforms the algorithms with which it is compared on the two proposed problems and CTPs, in terms of reliability and stability in finding a set of well-distributed optimal solutions.

Many-objective optimization problems (MaOPs) bring great hardship to most of the existing classical evolutionary multi-objective optimization algorithms, respecting selection operators, computational cost, visualization of the high-dimensional Pareto front, and so on. Objective reduction can help mitigate such hardship by removing the redundant objectives in the original objective set to some extent. Objective reduction problem is inherently a multiobjective optimization problem because we aim to preserve the dominance structure by using as little objectives as possible. In this paper, we propose an objective reduction algorithm based on constrained evolutionary multiobjective algorithm. A novel local search is presented for finding the Pareto-front by searching the neighbor. We compared the proposed algorithm with three algorithms on eleven test instances. The experimental results show the proposed algorithm can obtain a stable performance without any parameter need to fine tune.

Recently, evolutionary multi-tasking (EMT) has been proposed as a new evolutionary search paradigm that op-timizes multiple problems simultaneously. Due to the knowledge transfer across optimization tasks occurs along the evolutionary search process, EMT has been demonstrated to outperform the traditional single-task evolutionary search algorithms on many complex optimization problems, such as multimodal continuous optimization problems, NP-hard combinatorial optimization problems, and constrained optimization problems. Today, EMT has attracted lots of attentions, and many EMT algorithms have been proposed in the literature. The explicit EMT algorithm (EEMTA) is a recent proposed new EMT algorithm. In contrast to most of existing EMT algorithms, which employ a single population using unified space and common search operators for solving multiple problems, the EEMTA uses multiple populations which possess problem-specific solution representations and search mechanisms for different problems in evolutionary multi-tasking, which thus could lead to enhanced optimization performance. However, the original EEMTA was proposed for solving only two tasks. As knowledge transfer from inappropriate tasks may lead to negative effect on the evolutionary optimization process, additional designs of identifying task pairs for knowledge transfer is necessary in EEMTA for evolutionary multi-tasking with tasks more than two. To the best of our knowledge, there is no research effort has been conducted on this issue. Keeping this in mind, in this paper, we present a preliminary study on the task selection in EEMTA for many-task optimization. As task similarity may lose to capture the usefulness between tasks in evolutionary search, instead of using similarity measures for task selection, here we propose a credit assignment approach for selecting proper task to conduct knowledge transfer in explicit evolutionary many-tasking. The proposed approach is based on the feedbacks from the transferred solutions across tasks, which is adaptively updated along the evolutionary search. To confirm the efficacy of the proposed method, empirical studies on the many-task optimization problem, which consists of 7 commonly used optimization benchmarks, have been presented and discussed.

For solving constrained multi-objective optimization problems (CMOPs), an effective constraint-handling technique (CHT) is of great importance. Recently, many CHTs have been proposed for solving CMOPs. However, no single CHT can outperform all kinds of CMOPs. This paper proposes an algorithm, namely, ACHT-M2M, which adaptively allocates the existing CHTs in an M2M framework for solving CMOPs. To be more specific, a CMOP is first decomposed into several constrained multi-objective optimization subproblems by ACHT-M2M. Each subproblem has a subpopulation in a subregion. CHT for each subregion is adaptively allocated according to a proposed composite performance measure. Population for the next generation is selected from subregions by selection operators with different CHTs and the obtained nondominated feasible solutions in each generation are used to update a predefined archive. ACHT-M2M assembles the advantages of different CHTs and makes them cooperate with each other. The proposed ACHT-M2M is finally compared with the other 12 representative algorithms on benchmark CMOPs and the experimental results further confirm the effectiveness of ACHT-M2M for solving CMOPs.

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.

Despite the successful application of an extension of the Multi-Objective Evolution Algorithm based on Decomposition (MOEA/D-M2M) to solve unbalanced multi-objective optimization problems (UMOPs), its use in constrained unbalanced multi-objective optimization problems has not been fully explored. In an earlier paper, a definition of UMOPs was suggested that had two necessary conditions: 1) finding a favored subset of the Pareto set is easier than finding an unfavored subset, and 2) the favored subset of the Pareto set dominates a large part of the feasible space. The second condition strongly reduces the fraction of MOPs that are considered UMOPs. In this paper, we eliminate that second condition and consider a broader class of UMOPs. We design an unbalanced constrained multi-objective test suite with three different types of biased constraints, yielding three different types of constrained test problems in which the degree of imbalance is scalable via a set of parameters introduced for each problem. We analyse the characteristics of three types of constraints and the difficulties they present for potential solution algorithms–i.e., NSGA-II, MOEA/D and MOEA/D-M2M, with four constraint-handling techniques. MOEA/D-M2M is shown to significantly outperform the other algorithms on these problems due to its decomposition strategy.

The FIR digital filter design plays an important role in communication system. This paper proposes an optimal design method based on least squares method for the model of minimum mean square error criteria. This method derives the model's exact solution using least squares, and improves the solution accuracy. Meanwhile, the multi-objective evolutionary is used to optimize the weight value of sampling frequency to get excellent performance in the pass frequency band and stop frequency band. Experiments show the excellent performance of the FIR filter obtained by the proposed method.

Evolutionary Multi-objective optimization (EMO) Algorithms have been successfully applied in many practical problems. However, their performance deteriorated seriously for the problems with many objectives, i.e. many-objective optimization problems (MaOPs). In some practical applications, there exist some redundant objectives in an MaOP. Reducing the number of objectives of the optimization problem is one of the effective ways to solve the MaOPs with redundant objectives. It can improve the search efficiency of EMO algorithms, and reduce the computational cost. In this paper, we propose a new objective reduction algorithm. A criterion based on the number of non-dominated solution paired is presented to measure the conflict degree between objectives. Furthermore, we develop a effective objective reduction algorithm using feature selection technique. We compared the proposed algorithm with the LPCA, NLMVUP-CA and $\delta$-MOSS algorithm in some benchmark problems, and the results show the effectiveness of the proposed algorithm.

In this paper, we propose an evolutionary multi-and many-objective optimization algorithm combining I_SDE + and region decomposition. It decomposes the objective space into a number of sub-regions by a set of direction vectors and independently calculates the indicator I_SDE + by using the corresponding direction vector in each subregion. Thus, the convergence direction of each sub-region is relatively adjusted. In this way, the proposed algorithm can adapt to various of Pareto Front shapes. The inferior individuals are eliminated according to the value of I_SDE + of each individual one by one. In the experiments, we compare the proposed algorithm with four evolutionary multi-and many-objective optimization algorithms on WFG series with different number of objectives. The result shows that the proposed algorithm promotes diversity and convergence.

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.

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.

Multiobjective evolutionary algorithm based on decomposition has made a great contribution to the field of evolutionary multiobjective optimization problem. The decomposition-based algorithms construct a number of scalar optimization subproblems by using a set of weight vectors, and optimize these subproblems simultaneously to approximate the Pareto front (PF). The weight vectors have a massive influence on the performance of the decomposition-based algorithm, especially for the multiobjective optimization problems (MOP) with a complex PF. To solve this, we propose a parameterless decomposition scheme to adjust the weight vectors automatically. Experiment results indicate that the proposed algorithm can obtain better uniformity solutions for the MOP with complex PF.

This paper considers the minimum-energy control problem for a class of time-varying systems with multiple state and input delays. First, a state transform matrix is presented. By using the transform matrix, a system with multiple state and input delays is converted to a formal equivalent one without delay. Then, the optimal problem of the novel system is solved by using the maximum principle. Analytical expressions of the optimal control law and optimal performance are given by two formulas with respect to the state transform matrix. At last, an algorithm is given to solve the analytical expression of the state transform matrix. Simulation results show that the design algorithm is efficient and easy to implement.

The decomposition-based multi-objective evolutionary algorithm, i.e. MOEA/D-M2M, has shown to be an efficient algorithm to solve unconstrained imbalanced multiobjective optimization problems. However, the use in constrained imbalanced multi-objective optimization problems has not been fully explored. In this paper, we study the factors that impact the constrained imbalanced multi-objective optimization problems. To begin with, a series of constrained imbalanced multi-objective optimization problems are constructed. Then three kinds of representative algorithms, i.e. NSGA-II, MOEA/D and MOEA/D-M2M, combined with the constraint domination principle respectively, are utilized to solve them. The experimental results demonstrate that MOEA/D-M2M works better than the other two compared algorithms on constrained imbalanced multi-objective optimization problems in terms of the reliability and stability of finding a set of well distributed non-domination solutions.

Device to device (D2D) is a promising technology for the next generation mobile communication technology, and attracts a lot of attentions. Since D2D technology allows devices to connect directly without the assistance of the base station (BS), D2D can effectively improve system throughput and decrease the energy consumption. In this paper, we propose a novel spectrum sharing scheme between D2D pair and cellular users to improve the system sum rate. In the proposed scheme, cellular users can share spectrum resources with different D2D pairs which can effectively improve the spectrum efficiency. Meanwhile, the transmit power of both the D2D pairs and BS should be careful optimized since the spectrum sharing scheme will also lead to co-channel interference. Then, both the resource block (RB) assignment and power allocation are considered at the same time, and we formulate the resource allocation problem as a mixed integer programming problem which is difficult to tackle. To solve the problem, in this paper, we adopt the Nr-PSO to solve the proposed problem. The Nr-PSO is a current evolutionary algorithm which can improve the local search ability. Simulation results show that the effectiveness of the algorithm.

Decomposition based evolutionary algorithms have achieved great success in solving many-objective optimization problems. However, the design of proper decomposition vectors is not an easy task, especially in high dimensional objective space. In this paper, we study how to better use these decomposition vectors. We first show that for any given decomposition vector, new dominance relationship and crowding measurement strategy can be well defined. Based on them, we then propose a new evolutionary algorithm for many objective optimization. By this way, the utilization efficiency of decomposition vectors is enhanced and thus the task of weights design is alleviated accordingly. Experiments are conducted to compare the proposed algorithm with four state-of-the-art decomposition based evolutionary algorithms on a set of well-known many-objective test problems with 5 to 10 objectives. The simulation results show that the proposed algorithm can achieve comparable results with fewer decomposition vectors.

The performance of the most existing classical evolutionary multiobjective optimization (EMO) algorithms, especially for Pareto-based EMO algorithms, generally deteriorates over the number of objectives in solving many-objective optimization problems (MaOPs), in which the number of objectives is greater than three. Objective reduction methods that transform an MaOP into the one with few objectives, are a promising way for solving MaOPs. The dominance-based objective reduction methods, e.g. k-EMOSS and \(\delta \)-MOSS, omitting an objective while preserving the dominant structure of the individuals as much as possible, can achieve good performance. However, these algorithms have higher computational complexity. Therefore, this paper presents a novel measure for measuring the capacity of preserving the dominance structure of an objective set, i.e., the redundancy of an objective to an objective set. Subsequently, we propose a fast algorithm to find a minimum set of objectives preserving the dominance structure as much as possible. We compare the proposed algorithm with its counterparts on eleven test instances. Numerical studies show the effectiveness of the proposed algorithm.

In this paper, we present a new method to fast approximate the hypervolume measurement by improving the classical Monte Carlo sampling method. Hypervolume value can be used as a quality indicator or selection indicator for multiobjective evolutionary algorithms (MOEAs), and thus the efficiency of calculating this measurement is of crucial importance especially in the case of large sets or many dimensional objective spaces. To fast calculate hypervolume, we develop a new Monte Carlo sampling method by decreasing the amount of Monte Carlo sample points using a novel decomposition strategy in this paper. We first analyze the complexity of the proposed algorithm in theory, and then execute a series experiments to further test its efficiency. Both simulation experiments and theoretical analysis verify the effectiveness and efficiency of the proposed method.

The evolutionary multiobjective algorithms have been demonstrated the effectiveness in dealing with multiobjective optimization problems. However, when solving the problems with many objectives, i.e., the number of objectives is greater than three, it needs a large population size to maintain population diversity and provide a good approximation to the Pareto front. The dilemma between limited computational resources and the exponentially increasing population size is a big challenge. Thus, we suggest a hierarchical decomposition-based evolutionary algorithm for solving many-objective optimization problems in this paper. Specifically, it constructs a binary tree on a set of large-scale uniform weight vectors. We only compare a candidate solutions with the solutions on the path from root to a leaf node of the tree to assign it into an appropriate node. The proposed algorithm has lower time complexity. Theoretical analysis shows the complexity of the proposed algorithm is \(\mathcal {O}(Mlog(\mathbb {N}))\) for dealing with a new candidate solution. Empirical results fully demonstrate the effectiveness and competitiveness of the proposed algorithm.

Green communication has become a hot topic in the field of wireless communication. This paper aims to improve the Quality of Service (QoS) of the system and minimizes the energy consumption by the spectrum-energy cooperation between adjacent base stations. We formulate the proposed spectrum-energy cooperation model as a hybrid constrained many-objective optimization problem (MaOP). To improve the efficiency of optimization algorithm, an alternate optimization algorithm is presented to address the proposed complex MaOP. The evolutionary multiobjective algorithm is employed for spectrum cooperation optimization which is discrete optimization problem, meanwhile classical optimization method is employed for energy consumption optimization and energy cooperation optimization that are continuous optimization problems. Simulation results show the effectiveness of the algorithm.

Energy harvesting has drawn more and more attention because of the requirement of the green communication. It is well known that energy harvesting is one of the most important technologies of the green communication. Due to the intermittency of the renewable energy and the different space of the Base stations (BSs), the renewable energy cooperation is necessary to improve the renewable energy efficiency. In this paper, we propose a renewable energy cooperation scheme among different BSs, in which one BS can collect (share) energy from (to) another BSs. It is obvious that the energy cooperation efficiency will be improved when the number of the BSs that join the cooperation scheme is large. In this paper, our objective is to minimize the networks delay of each BS. To minimize the networks delay, it is reasonable that each BS hopes to transfer its energy to another BSs as little as possible, and collects energy from another BSs as much as possible. Thus, the objective of each BS are conflicting. We formulate the joint optimization problem into a many-objective mixed integer optimization problem which is difficult to optimize. We adopt the evolutionary many-objective algorithm based on decomposition and reference distance to optimize the problem. Simulation results show that the effectiveness of the algorithm.

In recent years, many constraint-handling techniques maintaining infeasible individuals in the population have demonstrated their effectiveness for solving constrained optimization problems. Generally, the effective utilization of the potential of the infeasible individuals will improve the performance of the algorithms. This paper proposes an approach based on decomposition for constrained multi-objective optimization problems. By transforming the degree of constraint violation into one objective, a constraint-handling technique is introduced to exploit useful infeasible individuals during the evolution process. Besides, along with the evolution, it puts more emphasis on infeasible individuals with better objective values and smaller constraint violations, since they can provide more importance information for improving the convergence and diversity of the population. The performance of the proposed algorithm is evaluated by using 8 well-known test instances, i.e. CTP-series test instances. And the experimental results indicate that our algorithm is superior to the compared algorithm.

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.

When optimizing an multiobjective optimization problem, the evolution of population can be regarded as a approximation to the Pareto Front (PF). Motivated by this idea, we propose an adaptive region decomposition framework: MOEA/D-AM2M for the degenerated Many-Objective optimization problem (MaOP), where degenerated MaOP refers to the optimization problem with a degenerated PF in a subspace of the objective space. In this framework, a complex MaOP can be adaptively decomposed into a number of many-objective optimization subproblems, which is realized by the adaptively direction vectors design according to the present population's distribution. A new adaptive weight vectors design method based on this adaptive region decomposition is also proposed for selection in MOEA/D-AM2M. This strategy can timely adjust the regions and weights according to the population's tendency in the evolutionary process, which serves as a remedy for the inefficiency of fixed and evenly distributed weights when solving MaOP with a degenerated PF. Five degenerated MaOPs with disconnected PFs are generated to identify the effectiveness of proposed MOEA/D-AM2M. Contrast experiments are conducted by optimizing those MaOPs using MOEA/D-AM2M, MOEA/D-DE and MOEA/D-M2M. Simulation results have shown that the proposed MOEA/D-AM2M outperforms MOEA/D-DE and MOEA/D-M2M.

An evolutionary multi-objective optimization (EMO) algorithm based on Pareto dominance have made great achievements in dealing with two or three objectives optimization problems. Nevertheless, with the problem objective function of the number increasing, the selection pressure of Pareto dominance based EMO algorithms deteriorates rapidly. In our paper, we propose an evolutionary algorithm based on M2M population decomposition and reference distance to solve many-objective optimization problems, named EAPD-RD. By population decomposition, the population is decomposed into several subpopulation, and each subpopulation corresponds to a subspace represented by a direction vector. Each individual has a projector distance along the direction vector and a reference distance to the direction vector. The proposed algorithm use projector distance to enhance selection pressure of Pareto dominance. A Niching techniques based on reference distance is designed to the maintain the population diversity. Contrast experiments have been conducted by testing DTLZ(1–4) problems with 8, 10 and 15 objectives with proposed algorithm, MOEA/D and GrEA. The results of simulation illustrate that our algorithm outperforms MOEA/D and GrEA under the IGD metric.

Research on multiobjective optimization problems becomes one of the hottest topics of intelligent computation. In order to improve the search efficiency of an evolutionary algorithm and maintain the diversity of solutions, in this paper, the learning automata (LA) is first used for quantization orthogonal crossover (QOX), and a new fitness function based on decomposition is proposed to achieve these two purposes. Based on these, an orthogonal evolutionary algorithm with LA for complex multiobjective optimization problems with continuous variables is proposed. The experimental results show that in continuous states, the proposed algorithm is able to achieve accurate Pareto-optimal sets and wide Pareto-optimal fronts efficiently. Moreover, the comparison with the several existing well-known algorithms: nondominated sorting genetic algorithm II, decomposition-based multiobjective evolutionary algorithm, decomposition-based multiobjective evolutionary algorithm with an ensemble of neighborhood sizes, multiobjective optimization by LA, and multiobjective immune algorithm with nondominated neighbor-based selection, on 15 multiobjective benchmark problems, shows that the proposed algorithm is able to find more accurate and evenly distributed Pareto-optimal fronts than the compared ones.

Despite the effectiveness of the decomposition-based multi-objective evolutional algorithm (MOEA/D-M2M) in solving continuous multi-objective optimization problems (MOPs), its performance in addressing 0/1 multi-objective knapsack problems (MOKPs) has not been fully explored. In this paper, we use MOEA/D-M2M with an improved greedy repair strategy to solve MOKPs. It first decomposes an MOKP into a number of simple optimization subproblems and solves them in a collaborative way. Each subproblem has its own subpopulation, and then an improved greedy strategy is introduced to improve the performance of the proposed algorithm on MOKPs. Therein, a weight vector chosen randomly from a corresponding subpopulation is utilized to repair infeasible individuals or improve feasible individuals to have a better fitness, which improves the convergence of the population. Experimental studies on a set of test instances indicate that the MOEA/D-M2M with the improved greedy strategy is superior to MOGLS and MOEA/D in terms of finding better approximations to the Pareto front.

Resource allocation for orthogonal frequency division multiplexing OFDM system, as a core technology for the 4th generation mobile communication system, has shown significant importance in the improvement of system transmission rate. At present, the two-step algorithm is described as a main method to deal with resource allocation for OFDM system. As carrier allocation and power allocation are not independent of each other, the two-step algorithm may result in lower complexity but poor convergence. This paper proposes a mixed evolutionary algorithm to allocate sub-carrier and power at the same time. It combines the simulated annealing algorithm and evolutionary algorithm. This paper also improves the definition of fairness in OFDM system. It can achieves perfect fairness among users while maintaining the efficient capacity performance. Simulation results demonstrate that the proposed resource allocation algorithm can achieve perfect performance, and increase the convergence rate greatly meanwhile efficiently improving the system capacity.

Evolutionary algorithm EA has been successfully applied to many numerical optimisation problems in recent years. However, EA has rather slow convergence rates on some optimisation problems. In this paper, a fast evolutionary algorithm FEA with searching preference is proposed. Our basic idea is that the better an individual is, the more resources are invested to search the region close to the individual. Two techniques are applied to achieve it: making the best individual found so far always take part in crossover and mutation, and proposing a novel crossover and mutation operator based on simulated annealing. Obviously, the search process emphasises the region around the best individual. Furthermore, we can prove that FEA converges to a global optimum in probability. Numerical simulations are conducted for 19 standard test functions. The performance of FEA is compared with three EAs FEP, OGA/Q and LEA that have been published recently. The results indicate that FEA is effective and efficient. Furthermore, the result obtained by FEP is better than the best result found so far for ƒ

There are two main tasks involved in addressing a multiobjective optimization problem (MOP) by evolutionary multiobjective (EMO) algorithms: 1) make the population converge close to the Pareto-optimal front and 2) maintain adequate population diversity. However, most state-of-the-art EMO algorithms are designed based on the “convergence first and diversity second” principle. It has been observed that although these EMO algorithms have been successful in optimizing many real-world MOPs, they fail to solve certain problems that feature a severe imbalance between diversity preservation and achieving convergence. This paper characterizes an imbalanced MOP by clearly defining properties and indicating the reasons for the existing EMO algorithms’ difficulties in solving them. We then present 14 imbalanced problems, with and without constraints. Computational results using four existing EMO algorithms—elitist non-dominated sorting genetic algorithm (NSGA-II), multiobjective evolutionary algorithm based on decomposition (MOEA/D), strength Pareto evolutionary algorithm 2 (SPEA2), and S metric selection EMO algorithm (SMS-EMOA) and a proposed generalized vector-evaluated genetic algorithm are then presented. It is seen that these EMO algorithms cannot solve these imbalanced problems, but they are able to solve the problems when augmented by multiobjective to multiobjective (M2M), an approach that decomposes the population into several interacting subpopulations. These results and the successful application of the EMO methods with the M2M approach even on standard so-called balanced problems indicate the usefulness of using the M2M approach.

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.