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To prevent the population from getting stuck in local areas and then missing the constrained Pareto front fragments in dealing with constrained multi-objective optimization problems (CMOPs), it is important to guide the population to evenly explore the promising areas that are not dominated by all examined feasible solutions. To this end, we first introduce a cost value based distance into the objective space, and then use this distance and the constraints to define an indicator to evaluate the contribution of each individual to exploring the promising areas. Theoretical studies show that the proposed indicator can effectively guide population to focus on exploring the promising areas without crowding in local areas. Accordingly, we propose a new constraint handling technique (CHT) based on this indicator. To further improve the diversity of population in the promising areas, the proposed indicator-based CHT divides the promising areas into multiple subregions, and then gives priority to removing the individuals with the worst fitness values in the densest subregions. We embed the indicator-based CHT in evolutionary algorithm and propose an indicator-based constrained multi-objective algorithm for solving CMOPs. Numerical experiments on several benchmark suites show the effectiveness of the proposed algorithm. Compared with six state-of-the-art constrained evolutionary multi-objective optimization algorithms, the proposed algorithm performs better in dealing with different types of CMOPs, especially in those problems that the individuals are easy to appear in the local infeasible areas that dominate the constrained Pareto front fragments.

Evolutionary algorithms (EAs) have been recognized as a promising approach for bi-level optimization. However, the population-based characteristic of EAs largely influences their efficiency and effectiveness due to the nested structure of the two levels of optimization problems. In this paper, we propose a transfer learning based parallel evolutionary algorithm (TLEA) framework for bi-level optimization. In this framework, the task of optimizing a set of lower level problems parameterized by upper level variables is conducted in a parallel manner. In the meanwhile, a transfer learning strategy is developed to improve the effectiveness of each lower level search (LLS) process. In practice, we implement two versions of the TLEA: the first version uses the covariance matrix adaptation evolutionary strategy and the second version uses the differential evolution as the evolutionary operator in lower level optimization. Experimental studies on two sets of widely used bi-level optimization benchmark problems are conducted, and the performance of the two TLEA implementations is compared to that of four well-established evolutionary bi-level optimization algorithms to verify the effectiveness and efficiency of the proposed algorithm framework.

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 ∊ + .

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.

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.

Over the past two decades, Evolutionary Multi-objective Optimization (EMO) algorithms have demonstrated their ability to find and maintain multiple trade-off solutions in two and three-objective problems, making EMO as one of the most emergent and exciting fields of research and application within Computational Intelligence (CI) area. The main reason for EMO's success is that the population-based EMO operators are able to establish an implicit parallel search within an evolving population to find multiple Pareto-optimal regions of the search space parallelly. For many-objective optimization problems involving a largedimensional objective space, the extent of implicit parallelism is argued here to be too generic, compared to the same in a lower-dimensional objective space. Decomposition-based EMO algorithms - a recent trend in EMO literature - which divide the overall computing task into a number of sub-tasks of focusing within a region of the search space have found to be successful in solving many-objective problems. In this paper, we study the effect of explicit control of an algorithm's implicit parallelism mechanism for achieving an enhanced performance of decomposition-based EMO algorithms. We consider three decomposition-based many-objective evolutionary algorithms (EAs) - MOEA/D, MOEA/D-M2M, and NSGA-III - for this purpose. We also investigate another explicit control strategy of suitably choosing a normalization method of objectives for improving the performance of MOEA/D and MOEA/ D-M2M methods, and report much improved performance than their original counterparts. The principles of this study are valid for any population-based search and optimization algorithms and can be extended to improve the performance other single-objective EA, EMO, and other relevant CI methods.

Evolutionary algorithms for bi-level optimization suffer from the low efficiency in dealing with the multiple lower level optimization tasks required by the population based upper level search. In this paper, we design a transfer learning based covariance matrix adaptation evolution strategy (TL-CMA-ES) for bi-level optimization problems, where the tasks of searching for multiple lower level optimal solutions are conducted by a set of CMA-ES optimizers in a parallel manner. Furthermore, a transfer learning strategy is introduced in the parallel lower level CMA-ES search such that each CMA-ES optimizer can learn and utilize useful features gained by its neighbours. Experimental comparison and analysis are carried out to verify the effectiveness and efficiency of the proposed method.

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.

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).

Numerous of multi-objective evolutionary algorithms have been presented in the past few years. However, most of existing evolutionary algorithms are aimed at the problem whose objectives are conflicting or related. And these algorithms fail to yield satisfactory results on the problems whose objectives are constituted of some independent objective subsets. This paper proposes a co-evolution algorithm for solving many-objective problems with independent objective sets. It decomposes a many-objective optimization problem into a set of subproblems according to relationships among the objectives. Each subproblem is optimized by a subpopulation, simultaneously. We compare the proposed algorithm with three algorithms NSGA-II, MOEA/D and HypE on six problems with independent objective sets. Experimental results have demonstrated that the proposed algorithm performs well on such problems with independent objective sets.

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.

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.

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.

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.

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.

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.

The wideband code division multiple access (WCDMA) network planning problem requires to determine the location and the configuration parameters of the base stations (BSs) so as to maximize the capacity and minimize the installation cost. This problem can be formulated as a complex set covering problem. Compared to the classical set covering problems, the coverage area of each BS is unknown in advance. This makes that the selection of each BS location and configuration parameters is determined by the location and configuration parameters of the neighbor BSs. Accordingly, we will conduct a competition and cooperation model based on the re-covered area of the BSs to measure the relationship of the BSs. Then, an efficient genetic operation based on this model is proposed to generate new-quality solutions. Further, four BS configuration parameters, i.e., the antenna height, antenna tilt, sector orientation and pilot signal power, are taken into account as well. Since there are too many combination levels of the configuration parameters, an encoding method based on orthogonal design is presented to reduce the search space. Subsequently, we merge the proposed encoding method and genetic operation into the multiobjective evolutionary algorithm-based decomposition (MOEA/D-M2M) to solve the WCDMA network planning problem. Simulation results show the efficacy of the proposed encoding and genetic operation in comparison with the existing counterpart.

There are several studies on hybrid multi-operator recombination methods, while few works have been proposed in the area of combining different fitness assignment in a framework. On the other hand, it is known that fitness assignment has a marked impact on the performance of evolutionary multiobjective optimization algorithm (EMOA). In this paper, a hybrid EMOA is proposed, which divides the population into several smaller subpopulations according to their distribution in the objective space. Each subpopulation is evolved by an individual EMOA, and a hybrid performance measure estimates the performance of these EMOAs. We focus on the fitness assignment and assume that all EMOAs used in the subpopulations adopt the same recombination operator. To evaluate performance of the proposed algorithm, we compare it with MOEA/D-M2M, MOE-A/D, SMS-EMOA and NSGA-II on 16 test instances. Experimental results show that the proposed algorithm performs better than or similar to those compared EMOAs.

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.

In this paper, we study the relay node placement problem of, given a deployment of sensor nodes, finding the minimum number of relay nodes to increase the lifetime and reduce the deployment costs of wireless sensor networks (WSNs) simultaneously. In particular, we use the average remaining energy of the relay nodes to express the lifetimes of the WSNs. Due to the conflict of the average remaining time and deployment costs of relay nodes, we first formulate the relay node placement problem as a multi-objective optimisation problem, which is difficult to address using traditional methods. Based on this, a novel evolutionary multi-objective algorithm is presented to solve the problem. Furthermore, a local search method is presented to improve the convergence of the algorithm. Simulation results show that the proposed relay node placement scheme can efficiently reduce the cost and increase the lifetime.

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.

Multi-objective evolutionary algorithms using the weighted sum of the objectives as the fitness functions feature simple execution and effectiveness in multi-objective optimization. However, they cannot fmd the Pareto solutions on the non-convex part of the Pareto frontier, and thus aze difficult to find evenly distributed solutions. Under the circumstances, this paper proposes anew evolutionary algorithm using multiple fitness functions. Although the weights generated via the sphere coordinate transformation and uniform design are used to define the fitness, the fitness is not defined by the weighted sum of the objectives. Instead, it is defined by the maximum value of the weighted normalized objectives using amin-max strategy. In this manner, the proposed algorithm can overcome the drawbacks of the algorithms using the weighted sum of the objectives, and explore the objective space to fmd approximate uniformly distributed solutions on the Pareto front gradually. The numerical simulations show t...

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.

This paper improves K-dominated sorting by M2M population decomposition [14] to enhance the population convergence and diversity, and propose a new evolutionary many objective optimization algorithm based on the improved Kdominated sorting. The proposed K-dominated sorting can maintain the population diversity by M2M population decomposition. What's more, we prove that the improved K-dominance can avoid circular dominance of the original K-dominance in theory. Compared with the Pareto-dominance, the improved K-dominance greatly increases the selection pressure of evolutionary algorithm when optimizing many-objective optimization problems. A new local density estimation method is used to improve population diversity. The contrast experiments of proposed algorithm and NSGA-II are conducted by optimizing the DTLZ series test functions. The simulation results show that the proposed algorithm has obvious advantages, which can not only improves the population convergence, but also ensure a good distribution along the Pareto-optimal Front.

Testing and assessing the performance of proposed algorithms in a right way is a very important part in evolutionary optimization. If the number of optimization objectives is more than three, it's commonly referred as many-objective optimization problem(MaOP). Despite the fact there exists many test functions and assessment metrics, all of which are designed for testing and assessing the performance of evolutionary algorithms, the test functions DTLZ proposed by Deb(2005) and the assessment metrics Inverted Generational Distance(IGD) proposed by Bosman and Thierens(2003) are both widely recognized and used in most of evolutionary optimizations, especially in many-objective evolutionary optimization. When using IGD as a assessment tool, the vital task is to distribute reference points uniformly along the Pareto front of test functions. To overcome it, however, is really difficult with the number of objectives increase. For dealing with such difficulty, this paper introduce an approach called Inverse Transform Technique(ITT) suggested by Fang and Wang(1999) and successfully distributes reference points uniformly along the Pareto front of DTLZ test functions. In the final, three representative evolutionary algorithms: NSGA-II, MOEA/D and MOEA/D-M2M are selected to test their performance in dealing with many-objective optimization. As a result, extensive experiments and analysis are conducted on four DTLZ test functions from five to ten objectives.

The excellent performance of evolutionary multi-objective algorithms based on S metric selection (SMS) has been identified by many researchers. However, huge computational effort of S metric calculation has limited the full application of those algorithms. This paper proposes a novel S metric selection evolutionary algorithm (SMS-M2M) based on the population decomposition strategy MOEA/D-M2M. In SMS-M2M, SMS is conducted in each subpopulation instead of the whole population, which can avoid the S metric calculation of the total population. The purpose of population decomposition is to directly reduce the huge computational effort of calculating S metric and thus to give a simple but effective method to improve the effectiveness of SMS based algorithm. SMS-M2M utilizes the same SMS with a popular SMS based evolutionary algorithm SMS-EMOA. Numerical studies of SMS-M2M and SMS-EMOA have shown that the M2M population decomposition can effectively reduce the computational effort of SMS, meanwhile the theoretic analysis identifies the efficiency and effectiveness of SMS-M2M.

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.

Incorporating decision maker's preference with multi objective optimization problems keeps an active research area. In this paper, we suggest an algorithm to integrate decision maker's preference into multiobjective evolutionary optimization algorithm based on decomposition technique (MOEA/D). Decomposition techniques require a set of evenly distributed weight vectors to generate a diverse set of solutions on the Pareto front. This newly proposed algorithm incorporates preferred weights generated by desirability functions. A set of evenly distributed weights in desirability space are mapped into objective space to represent decision maker's preference. The solutions corresponding to these preferred weights consist preferred population. Further, a second population associated with evenly distributed weights in objective space is utilized to boost the search for promising areas and present to the decision maker a global perspective of view. Experimental results show the algorithm could be able to find a set of trade-offs on the Pareto front.

In this paper, we propose a single objective optimization evolutionary algorithm (EA) based on Covariance Matrix Learning and Searching Preference (CMLSP) and design a switching method which is used to combine CMLSP and Covariance Matrix Adaptation Evolution Strategy (CMAES). Then we investigate the performance of the switch method on a set of 30 noiseless optimization problems designed for the special session on real-parameter optimization of CEC 2014. The basic idea of the proposed CMLSP is that it is more likely to find a better individual around a good individual. That is to say, the better an individual is, the more resources should be invested to search the region around the individual. To achieve it, we discard the traditional crossover and mutation and design a novel method based on the covariance matrix leaning to generate high quality solutions. The best individual found so far is used as the mean of a Gaussian distribution and the covariance of the best λ individuals in the population are used as the evaluation of its covariance matrix and we sample the next generation individual from the Gaussian distribution other than using crossover and mutation. In the process of generating new individuals, the best individual is changed if ever a better one is found. This search strategy emphasizes the region around the best individual so that a faster convergence can be achieved. The use of switch method is to make best use of the proposed CMLSP and existing CMAES. At last, we report the results.

It is well known that the performance of any evolutionary multiobjective optimization (EMO) algorithm over one class of problems is offset by the performance over another class by the “no free lunch” theorem. This means that there is no EMO algorithm can be regards as a panacea. Therefore, we propose an evolutionary multiobjective optimization algorithm with algorithm adaptive selection. It divides the population into several small subpopulations according to their distribution in the objective space. Each subpopulation owns a EMO algorithm, and make the worst agent on specific measures of performance learn from its neighbor best one according to the feedback from the search process. We test the proposed algorithm on nine widely used test instances. Experimental results have shown that the proposed algorithm is very competitive.

In Time Division Long Term EvolutionTD-LTE system, decreasing the cross time slot interference is very important to improve the system throughput. Accordingly, we will propose a multi-cells Orthogonal Frequency Division MultiplexingOFDM resource allocation model for TD-LTE system in this paper, which considers the cross time slot interference and the fairness. Subsequently, a TD-LTE resource allocation evolutionary algorithm based on adaptive fairness threshold is proposed to solve this model through introducing adaptive fairness threshold. Empirical studies show that the proposed model and algorithm can effectively decrease the cross time slot interference and improve the transmit rate of OFDM system under the same fairness.

With the improvement of complex and uncertain finance environment,the difficulty of portfolio problem is increasing. Whether or not the projects is successfully selected, directly affects the development of the investment companies. This paper firstly talks about the finance conditions in single term investment and then extends the investment from one term to many terms. After that, a multi-project and multi-term portfolio model through considering the remaining funds in different investment terms is proposed. The model is based on a new kind of Mean-Semi-covariance theory, which can describe the uncertainty of return and risk in investment. The portfolio investment is a multi-objective optimization problem with constraints. Multi-objective evolutionary algorithm (MOEA) with greedy repair strategy is used to deal with the infeasible individuals and makes the investment reasonable. Finally, computer simulation shows that the proposed algorithm can be considered as a viable alternative.

Planning of base stations (BS) is one of the fundamental problems in the fourth generation (4G) wireless network design. A new mathematical model for the 4G BS planning is proposed in this paper. The co-channel interference, orthogonal frequency division multiplexing (OFDM), the cell edge rate, the reference signal received power (RSRP) and the base station density are considered in detail in this model. The objectives of the model are to minimum construction cost, maximum coverage and capacity at the same time. It's a multiobjective optimization problem with constraints. So a multiobjective algorithm with a local search operator is designed to solve this model. Simulation results show that the according algorithm with local search operator can give a set of solutions with relatively lower cost, larger coverage, larger capacity and faster speed than the algorithm without local search operator. The effective and efficient of the proposed model can also be identified by the comparison of the solutions.

We propose a novel constrained multiobjective evolutionary algorithm based on decomposition and temporary register in this paper. It decomposes the constrained multiobjective optimization problem into a number of subproblems and then optimizes each subproblem in a collaborative way. We also propose a novel constraint handling technique based on temporary register. Each subproblem has its own subpopulation and one temporary register. The subpopulation is composed of those individuals which has better objective values and lower constraint violations of this subproblem, while the temporary register is composed of those individuals that are found before. We perform the crossover operator between each individual in the subpopulations and an individual which is randomly chosen from the corresponding temporary register. Therefore, the temporary register strategy makes the individuals which have better objective values and lower constraint violations have an opportunity to participate in the crossover and mutation, but don't been eliminated at once. Moreover, this constraint handling technique does not need any parameter setting. The numerical simulations show the proposed algorithm outperforms existing ones.

Orthogonal frequency division multiplexing (OFDM) resource allocation is a hybrid optimization problem that contains discrete variables and continuous variables. This paper proposes an evolutionary algorithm optimizing carrier allocation and power allocation at the same time. Since the proposed algorithm combines Kuhn-tucker conditions (KKT) conditions with evolutionary algorithm, the power allocation optimization can take advantage of the information of carrier allocation optimization, and the carrier allocation optimization can also take advantage of the information of power allocation optimization. Hence, the information of power allocation and carrier allocation can transfer each other, and the efficiency of the algorithm is improved. Numerical experiments show that the proposed algorithm can obviously improve the system capacity.

As an important constituent of wireless network planning, location area planning (LAP) directly affects the stability, security, and performance of wireless network. This work proposes a novel evolutionary algorithm (EA) to solve the LAP problem. The difference between the proposed algorithm and the previous EA is mainly how to encode. The new coding method is inspired by the famous four-color theorem in graph theory. Only four numbers are needed to encode all chromosomes by this method. The encoding and decoding process is fast and easy to implement. What is more, illegal solutions can be processed easily in the process of decoding. The design of effective and efficient genetic operators can also benefit from this coding method. The modified evolutionary algorithm with this coding method is especially effective for LAP problem. The use of the principle of fuzzy clustering in initialization can effectively compress the search space in this new algorithm. The computer simulation has been conducted, and the quality of proposed algorithm is confirmed by comparing the results of proposed algorithm with EA and simulated annealing (SA).