Convergence analysis of a self-adaptive multi-objective evolutionary algorithm based on grids

Evolutionary algorithms have been successfully applied to various multi-objective optimization problems. However, theoretical studies on multi-objective evolutionary algorithms, especially with self-adaption, are relatively scarce. This paper analyzes the convergence properties of a self-adaptive (μ+1)-algorithm. The convergence of the algorithm is defined, and general convergence conditions are studied. Under these conditions, it is proven that the proposed self-adaptive (μ+1)-algorithm converges in probability or almost surely to the Pareto-optimal front.     

Convergence performance comparison of quantum-inspired multi-objective evolutionary algorithms

In recent research, we proposed a general framework of quantum-inspired multi-objective evolutionary algorithms (QMOEA) and gave one of its sufficient convergence conditions to the Pareto optimal set. In this paper, two Q-gate operators, H gate and R&N gate, are experimentally validated as two Q-gate paradigms meeting the convergence condition. The former is a modified rotation gate, and the latter is a combination of rotation gate and NOT gate with the specified probability. To investigate their effectiveness and applicability, several experiments on the multi-objective 0/1 knapsack problems are carried out. Compared to two typical evolutionary algorithms and the QMOEA only with rotation gate, the QMOEA with H gate and R&N gate have more powerful convergence ability in high complex instances. Moreover, the QMOEA with R&N gate has the best convergence in almost all of the experimental problems. Furthermore, the appropriate ε value regions for two Q-gates are verified.     

混合分解和强度帕累托多目标进化算法

在多目标进化优化中,使用分解策略的基于分解的多目标进化算法(MOEA/D)时间复杂度低,使用〖BP(〗强度帕累托策略的〖BP)〗强度帕累托进化算法-2(SPEA2)能得到分布均匀的解集。结合这两种策略,提出一种新的多目标进化算法用于求解具有复杂、不连续的帕累托前沿的多目标优化问题(MOP)。首先,利用分解策略快速逼近帕累托前沿;然后,利用强度帕累托策略使解集均匀分布在帕累托前沿,利用解集重置分解策略中的权重向量集,使其适配于特定的帕累托前沿;最后,利用分解策略进一步逼近帕累托前沿。使用的反向世代距离(IGD)作为度量标准,将新算法与MOEA/D、SPEA2和paλ-MOEA/D在12个基准问题上进行性能对比。实验结果表明该算法性能在7个基准问题上最优,在5个基准问题上接近于最优,且无论MOP的帕累托前沿是简单或复杂、连续或不连续的,该算法均能生成分布均匀的解集。     

The assessment of evolutionary algorithms for analyzing the positional accuracy and uncertainty of maps

Pre-geodetic maps are an important part of our cultural heritage and a potential source of information for historical studies. Historical cartography should be evaluated in terms of precision and uncertainty prior to their use in any application. In the last decade, the majority of papers that address multi-objective optimization employed the concept of Pareto optimality. The goal of Pareto-based multi-objective strategies is to generate a front (set) of nondominated solutions as an approximation to the true Pareto-optimal front. This article proposes a solution for the problems of multi-objective accuracy and uncertainty analysis of pre-geodetic maps using four Pareto-based multi-objective evolutionary algorithms: HVSEA, NSGAII, SPEAII and msPESA. “The Geographic Atlas of Spain (AGE)” by Tomas Lopez in 1804 provides the cartography for this study. The results obtained from the data collected from the kingdoms of Extremadura and Aragon, sheets of maps (54-55-56-57) and (70-71-72-73), respectively, demonstrate the advantages of these multi-objective approaches compared with classical methods. The results show that the removal of 8% of the towns it is possible to obtain improvements of approximately 30% for HVSEA, msPESA and NSGAII. The comparison of these algorithms indicates that the majority of nondominated solutions obtained by NSGAII dominate the solutions obtained by msPESA and HVSEA; however, msPESA and HVSEA obtain acceptable extreme solutions in some instances. The Pareto fronts based on multi-objective evolutionary algorithms (MOEAs) are a better alternative when the uncertainty of map analyzed is high or unknown. Consequently, Pareto-based multi-objective evolutionary algorithms establish new perspectives for analyzing the positional accuracy and uncertainty of maps.     

A multi-objective evolutionary algorithm to exploit the similarities of resource allocation problems

The complexity of a resource allocation problem (RAP) is usually NP-complete, which makes an exact method inadequate to handle RAPs, and encourages heuristic techniques to this class of problems for obtaining approximate solutions in polynomial time. Different heuristic techniques have already been investigated for handling various RAPs. However, since the properties of an RAP can help in characterizing other RAPs, instead of individual solution techniques, the similarities of different RAPs might be exploited for developing a common solution technique for them. Two RAPs of quite different nature, namely university class timetabling and land-use management, are considered here for such a study. The similarities between the problems are first explored, and then a common multi-objective evolutionary algorithm (a kind of heuristic techniques) for them is developed by exploiting those similarities. The algorithm is problem-dependent to some extent and can easily be extended to other similar RAPs. In the present work, the algorithm is applied to two real instances of the considered problems, and its properties are derived from the obtained results.     

Wind turbine selection for wind farm layout using multi-objective evolutionary algorithms

Wind energy has become the world’s fastest growing energy source. Although wind farm layout is a well known problem, its solution used to be heuristic, mainly based on the designer experience. A key in search trend is to increase power production capacity over time. Furthermore the production of wind energy often involves uncertainties due to the stochastic nature of wind speeds. The addressed problem contains a novel aspect with respect of other wind turbine selection problems in the context of wind farm design. The problem requires selecting two different wind turbine models (from a list of 26 items available) to minimize the standard deviation of the energy produced throughout the day while maximizing the total energy produced by the wind farm. The novelty of this new approach is based on the fact that wind farms are usually built using a single model of wind turbine. This paper describes the usage of multi-objective evolutionary algorithms (MOEAs) in the context of power energy production, selecting a combination of two different models of wind turbine along with wind speeds distributed over different time spans of the day. Several MOEAs variants belonging to the most renowned and widely used algorithms such as SPEA2 NSGAII, PESA and msPEA have been investigated, tested and compared based on the data gathered from Cancun (Mexico) throughout the year of 2008. We have demonstrated the powerful of MOEAs applied to wind turbine selection problem (WTS) and estimate the mean power and the associated standard deviation considering the wind speed and the dynamics of the power curve of the turbines. Among them, the performance of PESA algorithm looks a little bit superior than the other three algorithms. In conclusion, the use of MOEAs is technically feasible and opens new perspectives for assisting utility companies in developing wind farms.     

Multi-objective evolutionary algorithms for energy-aware scheduling on distributed computing systems

The ongoing increase of energy consumption by IT infrastructures forces data center managers to find innovative ways to improve energy efficiency. The latter is also a focal point for different branches of computer science due to its financial, ecological, political, and technical consequences. One of the answers is given by scheduling combined with dynamic voltage scaling technique to optimize the energy consumption. The way of reasoning is based on the link between current semiconductor technologies and energy state management of processors, where sacrificing the performance can save energy.This paper is devoted to investigate and solve the multi-objective precedence constrained application scheduling problem on a distributed computing system, and it has two main aims: the creation of general algorithms to solve the problem and the examination of the problem by means of the thorough analysis of the results returned by the algorithms.The first aim was achieved in two steps: adaptation of state-of-the-art multi-objective evolutionary algorithms by designing new operators and their validation in terms of performance and energy. The second aim was accomplished by performing an extensive number of algorithms executions on a large and diverse benchmark and the further analysis of performance among the proposed algorithms. Finally, the study proves the validity of the proposed method, points out the best-compared multi-objective algorithm schema, and the most important factors for the algorithms performance.     

基于遗传算法求解多目标优化问题Pareto前沿

该文给出了传统的求解多目标优化方法存在的问题,引入了当前研究多目标优化的新方法———基于遗传算法求解问题的pareto解,讨论了该方法要解决的关键问题———多样性保持及解决策略,并给出了一个求解pareto解集的新算法,算法简单、高效、鲁棒性强。最后给出了实验结果。     

Solving a dial-a-ride problem with a hybrid evolutionary multi-objective approach: Application to demand responsive transport

Demand responsive transport allows customers to be carried to their destination as with a taxi service, provided that the customers are grouped in the same vehicles in order to reduce operational costs. This kind of service is related to the dial-a-ride problem. However, in order to improve the quality of service, demand responsive transport needs more flexibility. This paper tries to address this issue by proposing an original evolutionary approach. In order to propose a set of compromise solutions to the decision-maker, this approach optimizes three objectives concurrently. Moreover, in order to intensify the search process, this multi-objective evolutionary approach is hybridized with a local search. Results obtained on random and realistic problems are detailed to compare three state-of-the-art algorithms and discussed from an operational point of view.     

Illustration of fairness in evolutionary multi-objective optimization

It is widely assumed that evolutionary algorithms for multi-objective optimization problems should use certain mechanisms to achieve a good spread over the Pareto front. In this paper, we examine such mechanisms from a theoretical point of view and analyze simple algorithms incorporating the concept of fairness. This mechanism tries to balance the number of offspring of all individuals in the current population. We rigorously analyze the runtime behavior of different fairness mechanisms and present illustrative examples to point out situations, where the right mechanism can speed up the optimization process significantly. We also indicate drawbacks for the use of fairness by presenting instances, where the optimization process is slowed down drastically.     

A multi-objective evolutionary algorithm for examination timetabling

This paper considers the scheduling of exams for a set of university courses. The solution to this exam timetabling problem involves the optimization of complete timetables such that there are as few occurrences of students having to take exams in consecutive periods as possible but at the same time minimizing the timetable length and satisfying hard constraints such as seating capacity and no overlapping exams. To solve such a multi-objective combinatorial optimization problem, this paper presents a multi-objective evolutionary algorithm that uses a variable-length chromosome representation and incorporates a micro-genetic algorithm and a hill-climber for local exploitation and a goal-based Pareto ranking scheme for assigning the relative strength of solutions. It also imports several features from the research on the graph coloring problem. The proposed algorithm is shown to be a more general exam timetabling problem solver in that it does not require any prior information of the timetable length to be effective. It is also tested against a few influential and recent optimization techniques and is found to be superior on four out of seven publicly available datasets.     

A mathematical framework for passenger screening optimization via a multi-objective evolutionary approach

Concerns regarding the smuggling of dangerous items into commercial flights escalated after the failed Christmas day bomber attack. As a result, the Transportation Security Agency (TSA) has strengthened its efforts to detect passengers carrying hazardous items by installing novel screening technologies and by increasing the number of random pat-downs performed at security checkpoints nationwide. However, the implementation of such measures has raised privacy and health concerns among different groups thus making the design and evaluation of new inspection strategies strongly necessary. This research presents a mathematical framework to design passenger inspection strategies that include the utilization of novel and traditional technologies (i.e. body scanners, explosive detection systems, explosive trace detectors, walk-through metal detectors, and wands) offered by multiple manufacturers, to identify three types of items: metallic, bulk explosives (i.e. plastic, liquids, gels), and traces of explosives. A multiple objective optimization model is proposed to optimize inspection security, inspection cost, and processing time; an evolutionary approach is used to solve the model. The result is a Pareto set of quasi-optimal solutions representing multiple inspection strategies. Each strategy is different in terms of: (1) configuration, (2) the screening technologies included, (3) threshold calibration, and consequently, (4) inspection security, inspection cost, and processing time.     

An improved multi-objective evolutionary algorithm for the vehicle routing problem with time windows

The vehicle routing problem with time windows is a complex combinatorial problem with many real-world applications in transportation and distribution logistics. Its main objective is to find the lowest distance set of routes to deliver goods, using a fleet of identical vehicles with restricted capacity, to customers with service time windows. However, there are other objectives, and having a range of solutions representing the trade-offs between objectives is crucial for many applications. Although previous research has used evolutionary methods for solving this problem, it has rarely concentrated on the optimization of more than one objective, and hardly ever explicitly considered the diversity of solutions. This paper proposes and analyzes a novel multi-objective evolutionary algorithm, which incorporates methods for measuring the similarity of solutions, to solve the multi-objective problem. The algorithm is applied to a standard benchmark problem set, showing that when the similarity measure is used appropriately, the diversity and quality of solutions is higher than when it is not used, and the algorithm achieves highly competitive results compared with previously published studies and those from a popular evolutionary multi-objective optimizer.     

A dominance tree and its application in evolutionary multi-objective optimization

Most contemporary multi-objective evolutionary algorithms (MOEAs) store and handle a population with a linear list, and this may impose high computational complexities on the comparisons of solutions and the fitness assignment processes. This paper presents a data structure for storing the whole population and their dominating information in MOEAs. This structure, called a Dominance Tree (DT), is a binary tree that can effectively and efficiently store three-valued relations (namely dominating, dominated or non-dominated) among vector values. This paper further demonstrates DT’s potential applications in evolutionary multi-objective optimization with two cases. The first case utilizes the DT to improve NSGA-II as a fitness assignment strategy. The second case demonstrates a DT-based MOEA (called a DTEA), which is designed by leveraging the favorable properties of the DT. The simulation results show that the DT-improved NSGA-II is significantly faster than NSGA-II. Meanwhile, DTEA is much faster than SPEA2, NSGA-II and an improved version of NSGA-II. On the other hand, in regard to converging to the Pareto optimal front and maintaining the diversity of solutions, DT-improved NSGA-II and DTEA are found to be competitive with NSGA-II and SPEA2.     

Seeding the initial population of multi-objective evolutionary algorithms: A computational study

Most experimental studies initialize the population of evolutionary algorithms with random genotypes. In practice, however, optimizers are typically seeded with good candidate solutions either previously known or created according to some problem-specific method. This seeding has been studied extensively for single-objective problems. For multi-objective problems, however, very little literature is available on the approaches to seeding and their individual benefits and disadvantages. In this article, we are trying to narrow this gap via a comprehensive computational study on common real-valued test functions. We investigate the effect of two seeding techniques for five algorithms on 48 optimization problems with 2, 3, 4, 6, and 8 objectives. We observe that some functions (e.g., DTLZ4 and the LZ family) benefit significantly from seeding, while others (e.g., WFG) profit less. The advantage of seeding also depends on the examined algorithm.     

Distributed evolutionary multi-objective mesh-partitioning algorithm for parallel finite element computations

Majority of the mesh-partitioning algorithms attempt to optimise the interprocessor communications, while balancing the computational load among the processors. However, it is desirable to simultaneously optimise the submesh aspect ratios in order to significantly improve the convergence characteristics of the domain decomposition based Preconditioned-conjugate-gradient algorithms, being used extensively in the state-of-the-art parallel finite element codes. Keeping this in view, a new distributed multi-objective mesh-partitioning algorithm using evolutionary computing techniques is proposed in this paper. Effectiveness of the proposed distributed mesh-partitioning algorithm is demonstrated by solving several unstructured meshes of practical-engineering problems and also benchmark problems.     

一种快速构造多目标Pareto非支配集的方法：选举法则*

基于Pareto的多目标优化问题是进化算法的一个重要研究方向,而如何构造Pareto非支配集则是提高算法效率的关键所在。通过对选举现象的观察,同时针对多目标个体之间的特性,提出了一种快速求解多目标Pareto非支配集的方法: 选举法则(election principle,EP),分析了其时间复杂度为O(rmN),并对其进行了正确性证明。因为种群中实际的非支配个体数m比进化群体规模N小,所以与同类方法相比,EP有更高的效率,并通过了实验验证。     

模糊作业时间的拆卸线平衡Pareto多目标优化

针对实际拆卸作业的复杂性,建立了考虑模糊作业时间的多目标拆卸线平衡问题的数学模型,提出了一种基于Pareto解集的多目标遗传模拟退火算法进行求解。改进了模拟退火操作的Metropolis准则,使其能够求解多目标优化问题。采用拥挤距离评价非劣解的优劣,保留了优秀个体,并通过精英选择策略,将非劣解作为遗传操作的个体,引导算法向最优方向收敛。基于25项拆卸任务算例,通过与现有的单目标人工蜂群算法进行对比,验证了所提算法的有效性和优越性。最后将该算法应用于某打印机拆卸线实例中,求得8种可选平衡方案,实现了求解结果的多样性。     

Memory-based adaptive partitioning (MAP) of search space for the enhancement of convergence in Pareto-based multi-objective evolutionary algorithms

A new algorithm, dubbed memory-based adaptive partitioning (MAP) of search space, which is intended to provide a better accuracy/speed ratio in the convergence of multi-objective evolutionary algorithms (MOEAs) is presented in this work. This algorithm works by performing an adaptive-probabilistic refinement of the search space, with no aggregation in objective space. This work investigated the integration of MAP within the state-of-the-art fast and elitist non-dominated sorting genetic algorithm (NSGAII). Considerable improvements in convergence were achieved, in terms of both speed and accuracy. Results are provided for several commonly used constrained and unconstrained benchmark problems, and comparisons are made with standalone NSGAII and hybrid NSGAII-efficient local search (eLS).     

Physical programming for preference driven evolutionary multi-objective optimization

Preference articulation in multi-objective optimization could be used to improve the pertinency of solutions in an approximated Pareto front. That is, computing the most interesting solutions from the designer's point of view in order to facilitate the Pareto front analysis and the selection of a design alternative. This articulation can be achieved in an a priori, progressive, or a posteriori manner. If it is used within an a priori frame, it could focus the optimization process toward the most promising areas of the Pareto front, saving computational resources and assuring a useful Pareto front approximation for the designer. In this work, a physical programming approach embedded in an evolutionary multi-objective optimization is presented as a tool for preference inclusion. The results presented and the algorithm developed validate the proposal as a potential tool for engineering design by means of evolutionary multi-objective optimization.