Real Pareto Front Research Articles

A many-objective evolutionary algorithm based on three states for solving many-objective optimization problem

In recent years, researchers have taken the many-objective optimization algorithm, which can optimize 5, 8, 10, 15, 20 objective functions simultaneously, as a new research topic. However, the current research on many-objective optimization technology also encounters some challenges. For example: Pareto resistance phenomenon, difficult diversity maintenance. Based on the above problems, this paper proposes a many-objective evolutionary algorithm based on three states (MOEA/TS). Firstly, a feature extraction operator is proposed. It can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Secondly, based on Pareto front layer, the concept of “individual importance degree” is proposed. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer. Then, a repulsion field method is proposed. The diversity of the population in the objective space is maintained by the repulsion field, so that the population can be evenly distributed on the real Pareto front. Finally, a new concurrent algorithm framework is designed. In the algorithm framework, the algorithm is divided into three states, and each state focuses on a specific task. The population can switch freely among these three states according to its own evolution. The MOEA/TS algorithm is compared with 7 advanced many-objective optimization algorithms. The experimental results show that the MOEA/TS algorithm is more competitive in many-objective optimization problems.

A Web Data Mining Algorithm Based on Manifold Distance for Mixed Data in Cloud Service Architecture

Due to the complex distribution of web data and frequent updates under the cloud service architecture, the existing methods for global consistency of data ignore the global consistency of distance measurement and the inability to obtain neighborhood information of data. To overcome these problems, we transform the multi-information goal and multi-user demand (constraint conditions) in web data mining into a constrained multi-objective optimization model and solve it by a constrained particle swarm multi-objective optimization algorithm. While we measure the distance between data by manifold distance. In order to make it easier for the constrained multi-objective particle swarm algorithm to solve different types of problems to find an effective solution set closer to the real Pareto front, a new manifold learning algorithm based on the constrained multi-objective particle swarm algorithm is built and used to solve this problem. Experiments results demonstrate that this can improve the service efficiency of cloud computing.

Surrogate-Assisted Multi-Objective Evolutionary Optimization With Pareto Front Model-Based Local Search Method.

Some local search methods have been incorporated into surrogate-assisted multi-objective evolutionary algorithms to accelerate the search toward the real Pareto front (PF). In this article, a PF model-based local search method is proposed to accelerate the exploration and exploitation of the PF. It first builds a predicted PF model with current nondominated solutions. Then, some sparse points in the predicted PF are selected to guide the search directions of the local search in order to promote the search of promising sparse areas. The approximation degree of the predicted and real PFs will influence the speed of the local search, while extreme points can significantly influence the shape of the PF. To accelerate the search progress, the optima of surrogate models are utilized to promote the progress of finding extreme points. The proposed local search method is incorporated into a surrogate-assisted multi-objective evolutionary algorithm. The proposed surrogate-assisted multi-objective evolutionary algorithm with the proposed local search method is tested with Zitzler-Deb-Thiele (ZDT), Deb-Thiele-Laummans-Zitzler (DTLZ), and MAF instances. The experimental results demonstrated the efficiency of the proposed local search method and the superiority of the proposed algorithm.

An ensemble forecasting system for short-term power load based on multi-objective optimizer and fuzzy granulation

As an irreplaceable power source, electricity is responsible for sustaining the national economy and social development, and the precondition for the power system’s stable operation is to perform an accurate short-term load forecast (STLF). However, with the increasing forms of social power consumption and the emergence of large-scale sustainable resources on the grid, which make STLF increasingly challenging as the power load exhibits greater stochasticity and instability. Therefore, a novel STLF system is developed in this paper, which incorporates data fuzzy granulation, a high-performance optimizer for integrating forecasting sequences, point and interval forecasts. Moreover, the performance tests of the optimization algorithm verify that our proposed optimizer can obtain more approximate solution sets to the real Pareto front and outshines the traditional optimization algorithm concerning convergence and diversity. Load data from three regions of Australia demonstrate that our developed system can remarkably contribute to the accuracy and stability of the STLF, and also quantify the volatility and uncertainty of the power load, which allows power workers to better capture the fluctuation interval of future loads and effectively enhance the flexibility of grid operation.

Multiobjective Resource Allocation for mmWave MEC Offloading Under Competition of Communication and Computing Tasks

Towards 6G networks, such as virtual reality (VR) applications, Industry 4.0 and automated driving, demand mobile edge computing (MEC) techniques to offload computing tasks to nearby servers, which however causes fierce competition with traditional communication services. On the other hand, by introducing millimeter wave (mmWave) communication, it can significantly improve the offloading capability of MEC, so that enabling low latency and high throughput. For this sake, this paper investigates the resource management for the offload transmission of mmWave MEC system, when considering the data transmission demands from both communication-oriented users (CM-UEs) and computing-oriented users (CP-UEs). In particular, the joint consideration of user pairing, beamwidth allocation and power allocation is formulated as a multi-objective problem (MOP), which includes minimizing the offloading delay of CP-UEs and maximizing the transmission rate of CM-UEs. By using -constraint approach, the MOP is converted into a single-objective optimization problem (SOP) without losing Pareto optimality, and then the three-stage iterative resource allocation algorithm is proposed. Our simulation results show that, the gap between Pareto front generated by three-stage iterative resource allocation algorithm and the real Pareto front less than 0.16%. Futher, the proposed algorithm with much lower complexity can achieve the performance similar to the benchmark scheme of NSGA-2, while significantly outperforms the other traditional schemes.

Objective space division-based hybrid evolutionary algorithm for handing overlapping solutions in combinatorial problems.

Overlapping solutions occur when more than one solution in the space of decisions maps to the same solution in the space of objectives. This situation threatens the exploration capacity of Multi-Objective Evolutionary Algorithms (MOEAs), preventing them from having a good diversity in their population. The influence of overlapping solutions is intensified on multi-objective combinatorial problems with a low number of objectives. This paper presents a hybrid MOEA for handling overlapping solutions that combines the classic NSGA-II with a strategy based on Objective Space Division (OSD). Basically, in each generation of the algorithm, the objective space is divided into several regions using the nadir solution calculated from the current generation solutions. Furthermore, the solutions in each region are classified into non-dominated fronts using different optimization strategies in each of them. This significantly enhances the achieved diversity of the approximate front of non-dominated solutions. The proposed algorithm (called NSGA-II/OSD) is tested on a classic Operations Research problem: the Multi-Objective Knapsack Problem (0-1 MOKP) with two objectives. Classic NSGA-II, MOEA/D and Global WASF-GA are used to compare the performance of NSGA-II/OSD. In the case of MOEA/D two different versions are implemented, each of them with a different strategy for specifying the reference point. These MOEA/D reference point strategies are thoroughly studied and new insights are provided. This paper analyses in depth the impact of overlapping solutions on MOEAs, studying the number of overlapping solutions, the number of solution repairs, the hypervolume metric, the attainment surfaces and the approximation to the real Pareto front, for different sizes of 0-1 MOKPs with two objectives. The proposed method offers very good performance when compared to the classic NSGA-II, MOEA/D and Global WASF-GA algorithms, all of them well-known in the literature.

A Multiobjective Particle Swarm Optimization Algorithm Based on Grid Technique and Multistrategy

When faced with complex optimization problems with multiple objectives and multiple variables, many multiobjective particle swarm algorithms are prone to premature convergence. To enhance the convergence and diversity of the multiobjective particle swarm algorithm, a multiobjective particle swarm optimization algorithm based on the grid technique and multistrategy (GTMSMOPSO) is proposed. The algorithm randomly uses one of two different evaluation index strategies (convergence evaluation index and distribution evaluation index) combined with the grid technique to enhance the diversity and convergence of the population and improve the probability of particles flying to the real Pareto front. A combination of grid technology and a mixed evaluation index strategy is used to maintain the external archive to avoid removing particles with better convergence based only on particle density, which leads to population degradation and affects the particle exploitation ability. At the same time, a variation operation is proposed to avoid rapid degradation of the population, which enhances the particle search capability. The simulation results show that the proposed algorithm has better convergence and distribution than CMOPSO, NSGAII, MOEAD, MOPSOCD, and NMPSO.

Adaptive-surrogate-based robust optimization of transonic natural laminar flow nacelle

Natural Laminar Flow (NLF) technology is very effective for reducing the skin friction drag of aircraft engine nacelle, but the aerodynamic performance of NLF nacelle is highly sensitive to uncertain working conditions. Therefore, it’s imperative to incorporate uncertainties into the design of NLF nacelle. In this study, for a robust optimization of NLF nacelle and for improving its efficiency, an adaptive-surrogate-based robust optimization strategy is established, which is an iterative optimization process where the surrogate model is updated to obtain the real Pareto front of multi-objective optimization problem. A case study is carried out to validate its feasibility and effectiveness. The results show that the optimization increases the favorable pressure gradient region and the volume ratio of the nacelle by increasing its lip radius and reducing its maximum diameter. And the aerodynamic robustness of the NLF nacelle is mainly determined by the lip radius, maximum diameter of nacelle and location of the maximum diameter. Compared to the initial nacelle, the optimized nacelle maintains a wide range of low drag and high laminar flow ratio in the disturbance space, which extends the average laminar flow region to 21.6% and facilitates a decrease of 1.98 counts in the average drag coefficient.

Surrogate-assisted multi-objective particle swarm optimization for the operation of CO2 capture using VPSA

Multi-objective evolutionary algorithms (MOEAs) have received increasing attention over the past few decades. However, when applying MOEAs to solve computationally expensive real-world applications, they are often criticized due to the large number of function evaluations required. To this end, we proposed a method, called MOPSONN-EGO, for dealing with expensive multi-objective problems. MOPSONN-EGO uses Kriging models to approximate the objective function of expensive problems. The recently proposed MOPSONN algorithm is used to search the landscape of the Kriging models. The cheap-to-evaluate expected improvement matrix is adopted to select infill samples to update the Kriging models. Empirical results on several benchmark problems show the competitive performance of MOPSONN-EGO. Moreover, MOPSONN-EGO is applied to solve the multi-objective optimization problem of the Vacuum pressure swing adsorption (VPSA) for carbon dioxide capture. The simulation results are validated against the expensive model using normalized root mean square error (NRMSE) and compared to the One-shot surrogate modeling strategy. The results reveal the significant efficiency of the proposed algorithm in approximating the real Pareto front of VPSA using only a limited number of detailed model evaluations. Moreover, the obtained best values for each objective with their corresponding decision variables values are reported.

Direct Tracking of the Pareto Front of a Multi-Objective Optimization Problem

In this paper, some methodologies aimed at the identification of the Pareto front of a multi-objective optimization problem are presented and applied. Three different approaches are presented: local sampling, Pareto front resampling and Normal Boundary Intersection (NBI). A first approximation of the Pareto front is obtained by a regular sampling of the design space, and then the Pareto front is improved and enriched using the other two above mentioned techniques. A detailed Pareto front is obtained for an optimization problem where algebraic objective functions are applied, also in comparison with standard techniques. Encouraging results are also obtained for two different ship design problems. The use of the algebraic functions allows for a comparison with the real Pareto front, correctly detected. The variety of the ship design problems allows for a generalization of the applicability of the methodology.

Genetic algorithm for biobjective optimization of indoor LTE femtocell deployment

SummaryThe optimization of indoor femtocell base stations (FBS) placement is a highly complex problem in which multiple performance figures can be considered. In this paper, we propose a biobjective optimization approach based on non‐dominated sorting genetic algorithm II (NSGA‐II) to solve the indoor FBS deployment problem in residential scenarios where a single cell is deployed within the coverage of a single dominant interfering external macro base station (MBS). The optimization problem is formulated with the objective of jointly minimizing outage probability and maximizing normalized downlink average throughput. Several conditions are simultaneously considered under the framework of Monte Carlo simulations, such as the effect of co‐channel interference in uplink and downlink, buildings with arbitrary layouts, realistic transmission schemes and antenna radiation patterns, random user distributions, and a well‐established empirical path loss model. The proposed algorithm, which determines the location and orientation of the FBS's antenna that provides a suitable compromise between the two objective functions, is evaluated in this paper. First, two different experiments are proposed to investigate the algorithm convergence; then, statistical analyses are formulated to characterize the relation between the values achieved for both objective functions; finally, the computational load is evaluated. Results indicate the convergence of the proposed method towards the real Pareto front and a significant reduction of the computational time with respect to a reference method based on exhaustive search, while pointing at new criteria for the selection of the objective functions.

NSGA-III algorithm with maximum ranking strategy for many-objective optimisation

In recent years, a non-dominated sorting genetic algorithm III (NSGA-III) based on decomposition strategy had been extensively studied. However, there are still problems of lower Pareto selection pressure and insufficient diversity maintenance mechanism. To address these problems, NSGAIII algorithm with maximum ranking strategy (NSGAIII-MR) is proposed. In this algorithm, the convergence and diversity distance are balanced by adaptive parameter settings to achieve better performance. The maximum ranking strategy exploits the perpendicular distance from the solution to the weight vector to increase Pareto selection pressure. Moreover, the diversity of population is maintained with the reference point strategy to guide the solutions closer to the real Pareto front. Comparing with NSGAIII, the NSGAIII-MR algorithm enhances selection pressure and has good convergence and diversity performance. Also, the performance of algorithm is verified by comparing with other state-of-the-art evolutionary algorithms on the benchmark problems and the NSGAIII-MR is competitive.

NSGA-III algorithm with maximum ranking strategy for many-objective optimisation

In recent years, a non-dominated sorting genetic algorithm III (NSGA-III) based on decomposition strategy had been extensively studied. However, there are still problems of lower Pareto selection pressure and insufficient diversity maintenance mechanism. To address these problems, NSGAIII algorithm with maximum ranking strategy (NSGAIII-MR) is proposed. In this algorithm, the convergence and diversity distance are balanced by adaptive parameter settings to achieve better performance. The maximum ranking strategy exploits the perpendicular distance from the solution to the weight vector to increase Pareto selection pressure. Moreover, the diversity of population is maintained with the reference point strategy to guide the solutions closer to the real Pareto front. Comparing with NSGAIII, the NSGAIII-MR algorithm enhances selection pressure and has good convergence and diversity performance. Also, the performance of algorithm is verified by comparing with other state-of-the-art evolutionary algorithms on the benchmark problems and the NSGAIII-MR is competitive.

An Improved Multi-Objective Quantum-Behaved Particle Swarm Optimization for Railway Freight Transportation Routing Design

With the development of railway transportation, the railway transportation enterprises expand their freight transportation from station-to-station transportation to door-to-door transportation, which makes the routing design more complicated. The existing classical optimization algorithms are difficult to meet the needs of practical applications. Therefore, the paper introduces an Improved Multi-objective Quantum-behaved Particle Swarm Optimization algorithm (IMOQPSO). Then based on the continuous coding for the Railway Freight Transportation Routing Design, the proposed improved algorithm was applied to solve the problem to verify the performance of algorithm. Finally, the paper compared the performance of Improved Multi-objective Quantum-behaved Particle Swarm Optimization algorithm with other four continuous multi-objective swarm intelligence algorithms. The results shown that the proposed algorithm obtained the best Pareto front which is closer to the real Pareto front of Railway Freight Transportation Routing Design. Hence, the proposed Improved Multi-objective Quantum-behaved Particle Swarm Optimization algorithm can provide support for the railway transport enterprises routing design decisions to some extent.

A Multi-Objective Chaotic Particle Swarm Optimization Algorithm Based on Improved Inertial Weights

In order to enhance the convergence and distribution of multi-objective particle swarm algorithm, an improved multi-objective particle swarm optimization algorithm was proposed. Linear decreasing inertia weight was used to update. The method can improve the deficiency that the algorithm falls into local optimal easily. The improved Logistic mapping was used to increase ergodicity of the particles. The method can expand the search scope. At the same time, the elite archiving mechanism and the mutation probability were introduced to increase the disturbance. The method can improve the local optimal. Compared with the real Pareto front, NSGA-II and MOEAD algorithm, the simulation shows that the algorithm proposed in the paper is effective.

A corner point-based algorithm to solve constrained multi-objective optimization problems

The search ability of an algorithm in terms of convergence and diversity can be improved with the help of corner points. A corner point-based algorithm (CPA) based on a differential evolution (DE) algorithm is proposed to solve constrained multi-objective optimization problems. The evolutionary algorithm consists of two stages. The first stage is to find corner points by the proposed method. The second stage is to approach the real Pareto front. A novel diversity and convergence mechanism is implemented in the second stage. The performance of the proposed algorithm is evaluated on nineteen test functions. Compared with the constrained handling techniques and latest optimization algorithms, the numerical results have indicated that the proposed algorithm is effective. At last, the algorithm is used to solve resource schedule in emergency management to further validate its effectiveness.

Application of Ant Colony Algorithm in Multi-objective Optimization Problems

In actual application and scientific research, multi-objective optimization is an extremely important research subject. In reality, many issues are related to the simultaneous optimization under multi-objective conditions. The research subject of multi-objective optimization is getting increasing attention. In order to better solve some nonlinear, restricted complex multi-objective optimization problems, based on the current studies of multi-objective optimization and evolutionary algorithm, this paper applies the ant colony algorithm to multi-objective optimization, and proves through experiments that multi-objective ant colony algorithm can converge the real Pareto front of the standard test function more quickly and accurately, and can also maintain the distributivity of the better solution.

Diversity Comparison of Pareto Front Approximations in Many-Objective Optimization

Diversity assessment of Pareto front approximations is an important issue in the stochastic multiobjective optimization community. Most of the diversity indicators in the literature were designed to work for any number of objectives of Pareto front approximations in principle, but in practice many of these indicators are infeasible or not workable when the number of objectives is large. In this paper, we propose a diversity comparison indicator (DCI) to assess the diversity of Pareto front approximations in many-objective optimization. DCI evaluates relative quality of different Pareto front approximations rather than provides an absolute measure of distribution for a single approximation. In DCI, all the concerned approximations are put into a grid environment so that there are some hyperboxes containing one or more solutions. The proposed indicator only considers the contribution of different approximations to nonempty hyperboxes. Therefore, the computational cost does not increase exponentially with the number of objectives. In fact, the implementation of DCI is of quadratic time complexity, which is fully independent of the number of divisions used in grid. Systematic experiments are conducted using three groups of artificial Pareto front approximations and seven groups of real Pareto front approximations with different numbers of objectives to verify the effectiveness of DCI. Moreover, a comparison with two diversity indicators used widely in many-objective optimization is made analytically and empirically. Finally, a parametric investigation reveals interesting insights of the division number in grid and also offers some suggested settings to the users with different preferences.

Multi objective particle swarm optimization algorithm for the design of efficient ATO speed profiles in metro lines

One of the strategies for the reduction of energy consumption in railways systems is to execute efficient drivings (eco-driving). This eco-driving is the speed profile that requires the minimum energy consumption without degrading commercial running times or passenger comfort. When the trains are equipped with Automatic Train Operation systems (ATO) additional difficulties are involved. Their particular features make it necessary to develop accurate models that optimize the combination of the ATO commands of each speed profile to be used by the traffic regulation system. These commands are transmitted to the train via encoded balises on the track with little channel capacity (bandwidth). Thus, only a few and discrete values of the commands can be sent and the solution space of every interstation is made up of a relatively small set of speed profiles. However, the new state-of-the-art of signalling technologies permit a better bandwidth resulting in an exponential solution space. This calls for new methods for the optimal design of the ATO speed profiles without an exhaustive simulation of all the combinations. A MOPSO algorithm (Multi Objective Particle Swarm Optimization) to obtain the consumption/time Pareto front based on the simulation of a train with a real ATO system is proposed. The algorithm is able even to take into account only the comfortable speed profiles of the solution space. The fitness of the Pareto front is verified by comparing it with a NSGA-II algorithm (non-dominated sorting genetic algorithm II) and with the real Pareto front. Further, it has been used to obtain the optimal speed profiles in a real line of the Madrid Underground.

A Variable Neighborhood MOEA/D for Multiobjective Test Task Scheduling Problem

Test task scheduling problem (TTSP) is a typical combinational optimization scheduling problem. This paper proposes a variable neighborhood MOEA/D (VNM) to solve the multiobjective TTSP. Two minimization objectives, the maximal completion time (makespan) and the mean workload, are considered together. In order to make solutions obtained more close to the real Pareto Front, variable neighborhood strategy is adopted. Variable neighborhood approach is proposed to render the crossover span reasonable. Additionally, because the search space of the TTSP is so large that many duplicate solutions and local optima will exist, the Starting Mutation is applied to prevent solutions from becoming trapped in local optima. It is proved that the solutions got by VNM can converge to the global optimum by using Markov Chain and Transition Matrix, respectively. The experiments of comparisons of VNM, MOEA/D, and CNSGA (chaotic nondominated sorting genetic algorithm) indicate that VNM performs better than the MOEA/D and the CNSGA in solving the TTSP. The results demonstrate that proposed algorithm VNM is an efficient approach to solve the multiobjective TTSP.