Evolutionary Multi/Many-Objective Optimisation via Bilevel Decomposition

Decomposition of a complex multi-objective optimisation problem(MOP)to multiple simple subMOPs,known as M2M for short,is an effective approach to multi-objective optimisation.However,M2M facilitates little communication/collaboration between subMOPs,which limits its use in complex optimisation scenarios.This paper extends the M2M framework to develop a unified algorithm for both multi-objective and manyobjective optimisation.Through bilevel decomposition,an MOP is divided into multiple subMOPs at upper level,each of which is further divided into a number of single-objective subproblems at lower level.Neighbouring subMOPs are allowed to share some subproblems so that the knowledge gained from solving one subMOP can be transferred to another,and eventually to all the subMOPs.The bilevel decomposition is readily combined with some new mating selection and population update strategies,leading to a high-performance algorithm that competes effectively against a number of state-of-the-arts studied in this paper for both multiand many-objective optimisation.Parameter analysis and component analysis have been also carried out to further justify the proposed algorithm.

A Reference Vector-Assisted Many-Objective Optimization Algorithm with Adaptive Niche Dominance Relation

It is still a huge challenge for traditional Pareto-dominatedmany-objective optimization algorithms to solve manyobjective optimization problems because these algorithms hardly maintain the balance between convergence and diversity and can only find a group of solutions focused on a small area on the Pareto front,resulting in poor performance of those algorithms.For this reason,we propose a reference vector-assisted algorithmwith an adaptive niche dominance relation,for short MaOEA-AR.The new dominance relation forms a niche based on the angle between candidate solutions.By comparing these solutions,the solutionwith the best convergence is found to be the non-dominated solution to improve the selection pressure.In reproduction,a mutation strategy of k-bit crossover and hybrid mutation is used to generate high-quality offspring.On 23 test problems with up to 15-objective,we compared the proposed algorithm with five state-of-the-art algorithms.The experimental results verified that the proposed algorithm is competitive.

Many-Objective Optimization-Based Task Scheduling in Hybrid Cloud Environments

Due to the security and scalability features of hybrid cloud architecture,it can bettermeet the diverse requirements of users for cloud services.And a reasonable resource allocation solution is the key to adequately utilize the hybrid cloud.However,most previous studies have not comprehensively optimized the performance of hybrid cloud task scheduling,even ignoring the conflicts between its security privacy features and other requirements.Based on the above problems,a many-objective hybrid cloud task scheduling optimization model(HCTSO)is constructed combining risk rate,resource utilization,total cost,and task completion time.Meanwhile,an opposition-based learning knee point-driven many-objective evolutionary algorithm(OBL-KnEA)is proposed to improve the performance of model solving.The algorithm uses opposition-based learning to generate initial populations for faster convergence.Furthermore,a perturbation-based multipoint crossover operator and a dynamic range mutation operator are designed to extend the search range.By comparing the experiments with other excellent algorithms on HCTSO,OBL-KnEA achieves excellent results in terms of evaluation metrics,initial populations,and model optimization effects.

A Line Complex-Based Evolutionary Algorithm for Many-Objective Optimization

In solving many-objective optimization problems(MaO Ps),existing nondominated sorting-based multi-objective evolutionary algorithms suffer from the fast loss of selection pressure.Most candidate solutions become nondominated during the evolutionary process,thus leading to the failure of producing offspring toward Pareto-optimal front with diversity.Can we find a more effective way to select nondominated solutions and resolve this issue?To answer this critical question,this work proposes to evolve solutions through line complex rather than solution points in Euclidean space.First,Plücker coordinates are used to project solution points to line complex composed of position vectors and momentum ones.Besides position vectors of the solution points,momentum vectors are used to extend the comparability of nondominated solutions and enhance selection pressure.Then,a new distance function designed for high-dimensional space is proposed to replace Euclidean distance as a more effective distancebased estimator.Based on them,a novel many-objective evolutionary algorithm(MaOEA)is proposed by integrating a line complex-based environmental selection strategy into the NSGAⅢframework.The proposed algorithm is compared with the state of the art on widely used benchmark problems with up to 15 objectives.Experimental results demonstrate its superior competitiveness in solving MaOPs.

Design and optimization of diffraction-limited storage ring lattices based on many-objective evolutionary algorithms

Multi-objective evolutionary algorithms(MOEAs) are typically used to optimize two or three objectives in the accelerator field and perform well. However, the performance of these algorithms may severely deteriorate when the optimization objectives for an accelerator are equal to or greater than four. Recently, many-objective evolutionary algorithms(MaOEAs)that can solve problems with four or more optimization objectives have received extensive attention. In this study, two diffraction-limited storage ring(DLSR) lattices of the Extremely Brilliant Source(ESRF-EBS) type with different energies were designed and optimized using three MaOEAs and a widely used MOEA. The initial population was found to have a significant impact on the performance of the algorithms and was carefully studied. The performances of the four algorithms were compared, and the results demonstrated that the grid-based evolutionary algorithm(GrEA) had the best performance.Ma OEAs were applied in many-objective optimization of DLSR lattices for the first time, and lattices with natural emittances of 116 and 23 pm·rad were obtained at energies of 2 and 6 GeV, respectively, both with reasonable dynamic aperture and local momentum aperture(LMA). This work provides a valuable reference for future many-objective optimization of DLSRs.

MaOEA/I:Many-objective Evolutionary Algorithm Based on Indicator I_(ε+)

Balancing the diversity and convergence of the population is challenging in multi-objective optimization. The work proposed a many-objective evolutionary algorithm based on indicator I_(ε+)(MaOEA/I) to solve the above problems. Indicator I_(ε+)(x,y) is used for environmental selection to ensure diversity and convergence of the population. I_(ε+)(x,y) can evaluate the quality of individual x compared with individual y instead of the whole population. If I_(ε+)(x,y) is less than 0, individual x dominates y. If I_(ε+)(x,y) is 0, individuals x and y are the same. If I_(ε+)(x,y) is greater than 0, no dominant relationship exists between individuals x and y. The smaller I_(ε+)(x,y), the closer the two individuals. The dominated individuals should be deleted in environmental selection because they do not contribute to convergence. If there is no dominant individual, the same individuals and similar individuals should be deleted because they do not contribute to diversity. Therefore, the environmental selection of MaOEA/I should consider the two individuals with the smallest I_(ε+)(x,y). If I_(ε+)(x,y) is not greater than 0, delete individual y;if I_(ε+)(x,y) is greater than 0, check the distance between individuals x, y, and the target point and delete the individual with a longer distance. MaOEA/I is compared with 6 algorithms until the population does not exceed the population size. Experimental results demonstrate that MaOEA/I can gain highly competitive performance when solving many-objective optimization problems.

An ε-domination based two-archive 2 algorithm for many-objective optimization

The two-archive 2 algorithm(Two_Arch2) is a manyobjective evolutionary algorithm for balancing the convergence,diversity,and complexity using diversity archive(DA) and convergence archive(CA).However,the individuals in DA are selected based on the traditional Pareto dominance which decreases the selection pressure in the high-dimensional problems.The traditional algorithm even cannot converge due to the weak selection pressure.Meanwhile,Two_Arch2 adopts DA as the output of the algorithm which is hard to maintain diversity and coverage of the final solutions synchronously and increase the complexity of the algorithm.To increase the evolutionary pressure of the algorithm and improve distribution and convergence of the final solutions,an ε-domination based Two_Arch2 algorithm(ε-Two_Arch2) for many-objective problems(MaOPs) is proposed in this paper.In ε-Two_Arch2,to decrease the computational complexity and speed up the convergence,a novel evolutionary framework with a fast update strategy is proposed;to increase the selection pressure,ε-domination is assigned to update the individuals in DA;to guarantee the uniform distribution of the solution,a boundary protection strategy based on I_(ε+) indicator is designated as two steps selection strategies to update individuals in CA.To evaluate the performance of the proposed algorithm,a series of benchmark functions with different numbers of objectives is solved.The results demonstrate that the proposed method is competitive with the state-of-the-art multi-objective evolutionary algorithms and the efficiency of the algorithm is significantly improved compared with Two_Arch2.

Iterative learning-based many-objective history matching using deep neural network with stacked autoencoder

This paper presents an innovative data-integration that uses an iterative-learning method,a deep neural network(DNN)coupled with a stacked autoencoder(SAE)to solve issues encountered with many-objective history matching.The proposed method consists of a DNN-based inverse model with SAE-encoded static data and iterative updates of supervised-learning data are based on distance-based clustering schemes.DNN functions as an inverse model and results in encoded flattened data,while SAE,as a pre-trained neural network,successfully reduces dimensionality and reliably reconstructs geomodels.The iterative-learning method can improve the training data for DNN by showing the error reduction achieved with each iteration step.The proposed workflow shows the small mean absolute percentage error below 4%for all objective functions,while a typical multi-objective evolutionary algorithm fails to significantly reduce the initial population uncertainty.Iterative learning-based manyobjective history matching estimates the trends in water cuts that are not reliably included in dynamicdata matching.This confirms the proposed workflow constructs more plausible geo-models.The workflow would be a reliable alternative to overcome the less-convergent Pareto-based multi-objective evolutionary algorithm in the presence of geological uncertainty and varying objective functions.

A ε-indicator-based shuffled frog leaping algorithm for many-objective optimization problems

Many-objective optimization problems take challenges to multi-objective evolutionary algorithms.A number of nondominated solutions in population cause a difficult selection towards the Pareto front.To tackle this issue,a series of indicatorbased multi-objective evolutionary algorithms(MOEAs)have been proposed to guide the evolution progress and shown promising performance.This paper proposes an indicator-based manyobjective evolutionary algorithm calledε-indicator-based shuffled frog leaping algorithm(ε-MaOSFLA),which adopts the shuffled frog leaping algorithm as an evolutionary strategy and a simple and effectiveε-indicator as a fitness assignment scheme to press the population towards the Pareto front.Compared with four stateof-the-art MOEAs on several standard test problems with up to 50 objectives,the experimental results show thatε-MaOSFLA outperforms the competitors.

A Coevolutionary Algorithm for Many-Objective Optimization Problems with Independent and Harmonious Objectives

Evolutionary algorithm is an effective strategy for solving many-objective optimization problems.At present,most evolutionary many-objective algorithms are designed for solving many-objective optimization problems where the objectives conflict with each other.In some cases,however,the objectives are not always in conflict.It consists of multiple independent objective subsets and the relationship between objectives is unknown in advance.The classical evolutionary many-objective algorithms may not be able to effectively solve such problems.Accordingly,we propose an objective set decomposition strategy based on the partial set covering model.It decomposes the objectives into a collection of objective subsets to preserve the nondominance relationship as much as possible.An optimization subproblem is defined on each objective subset.A coevolutionary algorithm is presented to optimize all subproblems simultaneously,in which a nondominance ranking is presented to interact information among these sub-populations.The proposed algorithm is compared with five popular many-objective evolutionary algorithms and four objective set decomposition based evolutionary algorithms on a series of test problems.Numerical experiments demonstrate that the proposed algorithm can achieve promising results for the many-objective optimization problems with independent and harmonious objectives.

An Adaptive Many-objective Robust Optimization Model of Dynamic Reactive Power Sources for Voltage Stability Enhancement

The deployment of dynamic reactive power sourcecan effectively improve the voltage performance after a disturbance for a power system with increasing wind power penetration level and ubiquitous induction loads.To improve the voltage stability of the power system,this paper proposes an adaptive many-objective robust optimization model to deal with thedeployment issue of dynamic reactive power sources.Firstly,two metrics are adopted to assess the voltage stability of the system at two different stages,and one metric is proposed to assess the tie-line reactive power flow.Then,a robustness index isdeveloped to assess the sensitivity of a solution when subjectedto operational uncertainties,using the estimation of acceptablesensitivity region(ASR)and D-vine Copula.Five objectives areoptimized simultaneously:①total equipment investment;②adaptive short-term voltage stability evaluation;③tie-line power flow evaluation;④prioritized steady-state voltage stabilityevaluation;and⑤robustness evaluation.Finally,an anglebased adaptive many-objective evolutionary algorithm(MaOEA)is developed with two improvements designed for the application in a practical engineering problem:①adaptive mutationrate;and②elimination procedure without a requirement for athreshold value.The proposed model is verified on a modifiedNordic 74-bus system and a real-world power system.Numerical results demonstrate the effectiveness and efficiency of theproposed model.

Many-objective Optimization Method Based on Dimension Reduction for Operation of Large-scale Cooling Energy Systems

Large-scale cooling energy system has developed well in the past decade.However,its optimization is still a problem to be tackled due to the nonlinearity and large scale of existing systems.Reducing the scale of problems without oversimplifying the actual system model is a big challenge nowadays.This paper proposes a dimension reduction-based many-objective optimization(DRMO)method to solve an accurate nonlinear model of a practical large-scale cooling energy system.In the first stage,many-objective and many-variable of the large system are pre-processed to reduce the overall scale of the optimization problem.The relationships between many objectives are analyzed to find a few representative objectives.Key control variables are extracted to reduce the dimension of variables and the number of equality constraints.In the second stage,the manyobjective group search optimization(GSO)method is used to solve the low-dimensional nonlinear model,and a Pareto-front is obtained.In the final stage,candidate solutions along the Paretofront are graded on many-objective levels of system operators.The candidate solution with the highest average utility value is selected as the best running mode.Simulations are carried out on a 619-node-614-branch cooling system,and results show the ability of the proposed method in solving large-scale system operation problems.

Many-objective optimization for coordinated operation of integrated electricity and gas network

This paper develops a many-objective optimization model, which contains objectives representing the interests of the electricity and gas networks, as well as the distributed district heating and cooling units, to coordinate the benefits of all parties participated in the integrated energy system(IES). In order to solve the many-objective optimization model efficiently, an improved objective reduction(IOR) approach is proposed, aiming at acquiring the smallest set of objectives. The IOR approach utilizes the Spearman’s rank correlation coefficient to measure the relationship between objectives based on the Pareto-optimal front captured by the multi-objective group search optimizer with adaptive covariance and Lévy flights algorithm, and adopts various strategies to reduce the number of objectives gradually. Simulation studies are conducted on an IES consisting of a modified IEEE 30-bus electricity network and a 15-node gas network. The results show that the many-objective optimization problem is transformed into a bi-objective formulation by the IOR. Furthermore,our approach improves the overall quality of dispatch solutions and alleviates the decision making burden.

A many-objective evolutionary algorithm based on decomposition with dynamic resource allocation for irregular optimization

The multi-objective optimization problem has been encountered in numerous fields such as high-speed train head shape design,overlapping community detection,power dispatch,and unmanned aerial vehicle formation.To address such issues,current approaches focus mainly on problems with regular Pareto front rather than solving the irregular Pareto front.Considering this situation,we propose a many-objective evolutionary algorithm based on decomposition with dynamic resource allocation(Ma OEA/D-DRA)for irregular optimization.The proposed algorithm can dynamically allocate computing resources to different search areas according to different shapes of the problem’s Pareto front.An evolutionary population and an external archive are used in the search process,and information extracted from the external archive is used to guide the evolutionary population to different search regions.The evolutionary population evolves with the Tchebycheff approach to decompose a problem into several subproblems,and all the subproblems are optimized in a collaborative manner.The external archive is updated with the method of rithms using a variety of test problems with irregular Pareto front.Experimental results show that the proposed algorithèm out-p￡performs these five algorithms with respect to convergence speed and diversity of population members.By comparison with the weighted-sum approach and penalty-based boundary intersection approach,there is an improvement in performance after integration of the Tchebycheff approach into the proposed algorithm.

Reference direction based immune clone algorithm for many-objective optimization

In this paper, a new preference multi-objective optimization algorithm called immune clone algorithm based on reference direction method （RD-ICA） is proposed for solving many-objective optimization problems. First, an intelligent recombination operator, which performs well on the functions comprising many parameters, is introduced into an immune clone algorithm so as to explore the potentially excellent gene segments of all individuals in the antibody pop- ulation. Second, a reference direction method, a very strict ranking based on the desire of decision makers （DMs）, is used to guide selection and clone of the active population. Then a light beam search （LBS） is borrowed to pick out a small set of individuals filling the external population. The proposed method has been extensively compared with other recently proposed evolutionary multi-objective optimization （EMO） approaches over DTLZ problems with from 4 to 100 objectives. Experimental results indicate RD-ICA can achieve competitive results.

Multi-UAV Cooperative Trajectory Planning Based on Many-Objective Evolutionary Algorithm

The trajectory planning of multiple unmanned aerial vehicles(UAVs)is the core of efficient UAV mission execution.Existing studies have mainly transformed this problem into a single-objective optimization problem using a single metric to evaluate multi-UAV trajectory planning methods.However,multi-UAV trajectory planning evolves into a many-objective optimization problem due to the complexity of the demand and the environment.Therefore,a multi-UAV cooperative trajectory planning model based on many-objective optimization is proposed to optimize trajectory distance,trajectory time,trajectory threat,and trajectory coordination distance costs of UAVs.The NSGA-III algorithm,which overcomes the problems of traditional trajectory planning,is used to solve the model.This paper also designs a segmented crossover strategy and introduces dynamic crossover probability in the crossover operator to improve the solving efficiency of the model and accelerate the convergence speed of the algorithm.Experimental results prove the effectiveness of the multi-UAV cooperative trajectory planning algorithm,thereby addressing different actual needs.

Novel PIO Algorithm with Multiple Selection Strategies for Many-Objective Optimization Problems

With the increase of problem dimensions,most solutions of existing many-objective optimization algorithms are non-dominant.Therefore,the selection of individuals and the retention of elite individuals are important.Existing algorithms cannot provide sufficient solution precision and guarantee the diversity and convergence of solution sets when solving practical many-objective industrial problems.Thus,this work proposes an improved many-objective pigeon-inspired optimization(ImMAPIO)algorithm with multiple selection strategies to solve many-objective optimization problems.Multiple selection strategies integrating hypervolume,knee point,and vector angles are utilized to increase selection pressure to the true Pareto Front.Thus,the accuracy,convergence,and diversity of solutions are improved.ImMAPIO is applied to the DTLZ and WFG test functions with four to fifteen objectives and compared against NSGA-III,GrEA,MOEA/D,RVEA,and many-objective Pigeon-inspired optimization algorithm.Experimental results indicate the superiority of ImMAPIO on these test functions.

The Optimization of Manufacturing Resources Allocation Considering the Geographical Distribution

From the perspective of the geographical distribution, considering production fare, supply chain information and quality rating of the manufacturing resource(MR), a manufacturing resource allocation(MRA) model considering the geographical distribution in cloud manufacturing(CM) environment is built. The model includes two stages, preliminary selection stage and optimal selection stage. The membership function is used to select MRs from cloud resource pool(CRP) in the first stage, and then the candidate resource pool is built. In the optimal selection stage, a multi-objective optimization algorithm, particle swarm optimization(PSO) based on the method of relative entropy of fuzzy sets(REFS_PSO), is used to select optimal MRs from the candidate resource pool, and an optimal manufacturing resource supply chain is obtained at last. To verify the performance of REFS_PSO, NSGA-Ⅱ and PSO based on random weighting(RW_PSO) are selected as the comparison algorithms. They all are used to select optimal MRs at the second stage. The experimental results show solution obtained by REFS_PSO is the best. The model and the method proposed are appropriate for MRA in CM.

Multi-objective optimal dispatch of household flexible loads based on their real-life operating characteristics and energy-related occupant behavior

A model-based optimal dispatch framework was proposed to optimize operation of residential flexible loads considering their real-life operating characteristics,energy-related occupant behavior,and the benefits of different stakeholders.A pilot test was conducted for a typical household.According to the monitored appliance-level data,operating characteristics of flexible loads were identified and the models of these flexible loads were developed using multiple linear regression and K-means clustering methods.Moreover,a data-mining approach was developed to extract the occupant energy usage behavior of various flexible loads from the monitored data.Occupant behavior of appliance usage,such as daily turn-on times,turn-on moment,duration of each operation,preference of temperature setting,and flexibility window,were determined by the developed data-mining approach.Based on the established flexible load models and the identified occupant energy usage behavior,a many-objective nonlinear optimal dispatch model was developed aiming at minimizing daily electricity costs,occupants’dissatisfaction,CO_(2) emissions,and the average ramping index of household power profiles.The model was solved with the assistance of the NSGA-III and TOPSIS methods.Results indicate that the proposed framework can effectively optimize the operation of household flexible loads.Compared with the benchmark,the daily electricity costs,CO_(2) emissions,and average ramping index of household power profiles of the optimal plan were reduced by 7.3%,6.5%,and 14.4%,respectively,under the TOU tariff,while those were decreased by 9.5%,8.8%,and 23.8%,respectively,under the dynamic price tariff.The outputs of this work can offer guidance for the day-ahead optimal scheduling of household flexible loads in practice.

Search-Based Software Test Data Generation for Path Coverage Based on a Feedback-Directed Mechanism

Automatically generating test cases by evolutionary algorithms to satisfy the path coverage criterion has attracted much research attention in software testing.In the context of generating test cases to cover many target paths,the efficiency of existing methods needs to be further improved when infeasible or difficult paths exist in the program under test.This is because a significant amount of the search budget(i.e.,time allocated for the search to run)is consumed when computing fitness evaluations of individuals on infeasible or difficult paths.In this work,we present a feedback-directed mechanism that temporarily removes groups of paths from the target paths when no improvement is observed for these paths in subsequent generations.To fulfill this task,our strategy first organizes paths into groups.Then,in each generation,the objective scores of each individual for all paths in each group are summed up.For each group,the lowest value of the summed up objective scores among all individuals is assigned as the best aggregated score for a group.A group is removed when no improvement is observed in its best aggregated score over the last two generations.The experimental results show that the proposed approach can significantly improve path coverage rates for programs under test with infeasible or difficult paths in case of a limited search budget.In particular,the feedback-directed mechanism reduces wasting the search budget on infeasible paths or on difficult target paths that require many fitness evaluations before getting an improvement.