Achieving balance between proximity and diversity in multi-objective evolutionary algorithm

Currently, an alternative framework using the hypervolume indicator to guide the search for elite solutions of a multi-objective problem is studied in the evolutionary multi-objective optimization community very actively, comparing to the traditional Pareto dominance based approach. In this paper, we present a dynamic neighborhood multi-objective evolutionary algorithm based on hypervolume indicator (DNMOEA/HI), which benefits from both Pareto dominance and hypervolume indicator based frameworks. DNMOEA/HI is featured by the employment of hypervolume indicator as a truncation operator to prune the exceeded population, while a well-designed density estimator (i.e., tree neighborhood density) is combined with the Pareto strength value to perform fitness assignment. Moreover, a novel algorithm is proposed to directly evaluate the hypervolume contribution of a single individual. The performance of DNMOEA/HI is verified on a comprehensive benchmark suite, in comparison with six other multi-objective evolutionary algorithms. Experimental results demonstrate the efficiency of our proposed algorithm. Solutions obtained by DNMOEA/HI well approach the Pareto optimal front and are evenly distributed over the front, simultaneously.     

A diversity preserving selection in multiobjective evolutionary algorithms

In this paper, an efficient diversity preserving selection (DPS) technique is presented for multiobjective evolutionary algorithms (MEAs). The main goal is to preserve diversity of nondominated solutions in problems with scaled objectives. This is achieved with the help of a mechanism that preserves certain inferior individuals over successive generations with a view to provide long term advantages. The mechanism selects a group (of individuals) that is statistically furthest from the worst group, instead of just concentrating on the best individuals, as in truncation selection. In a way, DPS judiciously combines the diversity preserving mechanism with conventional truncation selection. Experiments demonstrate that DPS significantly improves diversity of nondominated solutions in badly-scaling problems, while at the same time it exhibits acceptable proximity performance. Whilst DPS has certain advantages when it comes to scaling problems, it empirically shows no disadvantages for the problems with non-scaled objectives.     

A tool for multiobjective evolutionary algorithms

This paper introduces a software tool based on illustrative applications for the development, analysis and application of multiobjective evolutionary algorithms. The multiobjective evolutionary algorithms tool (MOEAT) written in C# using a variety of multiobjective evolutionary algorithms (MOEAs) offers a powerful environment for various kinds of optimization tasks. It has many useful features such as visualizing of the progress and the results of optimization in a dynamic or static mode, and decision variable settings. The performance measurements of well-known multiobjective evolutionary algorithms in MOEAT are done using benchmark problems. In addition, two case studies from engineering domain are presented.     

An improved decomposition-based multiobjective evolutionary algorithm with a better balance of convergence and diversity

In decomposition-based multiobjective evolutionary algorithms (MOEAs), a good balance between convergence and diversity is very important to the performance of an algorithm. However, only the aggregation functions enough to achieve a good balance, especially in high-dimensional objective space. So we considered using the value of related acute angle between a solution and a direction vector as an other consider index. This idea is implemented to enhance the famous decomposition-based algorithm, i.e., MOEA/D. The enhanced algorithm is compared to its predecessor and other state-of-the-art algorithms on a several well-known test suites. Our experimental results show that the proposed algorithm performs better than its predecessor in keeping a better balance between the convergence and diversity, and also as effective as other state-of-the-art algorithms.     

Speeding up backpropagation using multiobjective evolutionary algorithms

The use of backpropagation for training artificial neural networks (ANNs) is usually associated with a long training process. The user needs to experiment with a number of network architectures; with larger networks, more computational cost in terms of training time is required. The objective of this letter is to present an optimization algorithm, comprising a multiobjective evolutionary algorithm and a gradient-based local search. In the rest of the letter, this is referred to as the memetic Pareto artificial neural network algorithm for training ANNs. The evolutionary approach is used to train the network and simultaneously optimize its architecture. The result is a set of networks, with each network in the set attempting to optimize both the training error and the architecture. We also present a self-adaptive version with lower computational cost. We show empirically that the proposed method is capable of reducing the training time compared to gradient-based techniques.     

基于Pareto最优概念的多目标进化算法研究

基于Pareto最优概念的多目标进化算法已成为多目标优化问题研究的主流方向。详细介绍了该领域的经典算法,重点阐述了各种算法在种群快速收敛并均匀分布于问题的非劣最优域上所采取的策略,并归纳了算法性能评估中需要进一步研究的几个问题。     

Combining convergence and diversity in evolutionary multiobjective optimization

Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to find a number of Pareto-optimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms can progress towards the Pareto-optimal set with a widely spread distribution of solutions. However, none of the multiobjective evolutionary algorithms (MOEAs) has a proof of convergence to the true Pareto-optimal solutions with a wide diversity among the solutions. In this paper, we discuss why a number of earlier MOEAs do not have such properties. Based on the concept of epsilon-dominance, new archiving strategies are proposed that overcome this fundamental problem and provably lead to MOEAs that have both the desired convergence and distribution properties. A number of modifications to the baseline algorithm are also suggested. The concept of epsilon-dominance introduced in this paper is practical and should make the proposed algorithms useful to researchers and practitioners alike.     

Comparison of multiobjective evolutionary algorithms: empirical results

In this paper, we provide a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions. Each test function involves a particular feature that is known to cause difficulty in the evolutionary optimization process, mainly in converging to the Pareto-optimal front (e.g., multimodality and deception). By investigating these different problem features separately, it is possible to predict the kind of problems to which a certain technique is or is not well suited. However, in contrast to what was suspected beforehand, the experimental results indicate a hierarchy of the algorithms under consideration. Furthermore, the emerging effects are evidence that the suggested test functions provide sufficient complexity to compare multiobjective optimizers. Finally, elitism is shown to be an important factor for improving evolutionary multiobjective search.     

Performance evaluation of evolutionary multiobjective optimization algorithms for multiobjective fuzzy genetics-based machine learning

Recently, evolutionary multiobjective optimization (EMO) algorithms have been utilized for the design of accurate and interpretable fuzzy rule-based systems. This research area is often referred to as multiobjective genetic fuzzy systems (MoGFS), where EMO algorithms are used to search for non-dominated fuzzy rule-based systems with respect to their accuracy and interpretability. In this paper, we examine the ability of EMO algorithms to efficiently search for Pareto optimal or near Pareto optimal fuzzy rule-based systems for classification problems. We use NSGA-II (elitist non-dominated sorting genetic algorithm), its variants, and MOEA/D (multiobjective evolutionary algorithm based on decomposition) in our multiobjective fuzzy genetics-based machine learning (MoFGBML) algorithm. Classification performance of obtained fuzzy rule-based systems by each EMO algorithm is evaluated for training data and test data under various settings of the available computation load and the granularity of fuzzy partitions. Experimental results in this paper suggest that reported classification performance of MoGFS in the literature can be further improved using more computation load, more efficient EMO algorithms, and/or more antecedent fuzzy sets from finer fuzzy partitions.     

MIJ2K Optimization using evolutionary multiobjective optimization algorithms

This paper deals with the multiobjective definition of video compression and its optimization. The optimization will be done using NSGA-II, a well-tested and highly accurate algorithm with a high convergence speed developed for solving multiobjective problems. Video compression is defined as a problem including two competing objectives. We try to find a set of optimal, so-called Pareto-optimal solutions, instead of a single optimal solution. The two competing objectives are quality and compression ratio maximization. The optimization will be achieved using a new patent pending codec, called MIJ2K, also outlined in this paper. Video will be compressed with the MIJ2K codec applied to some classical videos used for performance measurement, selected from the Xiph.org Foundation repository. The result of the optimization will be a set of near-optimal encoder parameters. We also present the convergence of NSGA-II with different encoder parameters and discuss the suitability of MOEAs as opposed to classical search-based techniques in this field.     

Mean-semivariance portfolio optimization with multiobjective evolutionary algorithms and technical analysis rules

Recent work has been devoted to study the use of multiobjective evolutionary algorithms (MOEAs) in stock portfolio optimization, within a common mean-variance framework. This article proposes the use of a more appropriate framework, mean-semivariance framework, which takes into account only adverse return variations instead of overall variations. It also proposes the use and comparison of established technical analysis (TA) indicators in pursuing better outcomes within the risk-return relation. Results show there is some difference in the performance of the two selected MOEAs – non-dominated sorting genetic algorithm II (NSGA II) and strength pareto evolutionary algorithm 2 (SPEA 2) – within portfolio optimization. In addition, when used with four TA based strategies – relative strength index (RSI), moving average convergence/divergence (MACD), contrarian bollinger bands (CBB) and bollinger bands (BB), the two selected MOEAs achieve solutions with interesting in-sample and out-of-sample outcomes for the BB strategy.     

Running time analysis of multiobjective evolutionary algorithms on pseudo-Boolean functions

This paper presents a rigorous running time analysis of evolutionary algorithms on pseudo-Boolean multiobjective optimization problems. We propose and analyze different population-based algorithms, the simple evolutionary multiobjective optimizer (SEMO), and two improved versions, fair evolutionary multiobjective optimizer (FEMO) and greedy evolutionary multiobjective optimizer (GEMO). The analysis is carried out on two biobjective model problems, leading ones trailing zeroes (LOTZ) and count ones count zeroes (COCZ), as well as on the scalable m-objective versions mLOTZ and mCOCZ. Results on the running time of the different population-based algorithms and for an alternative approach, a multistart (1+1)-EA based on the /spl epsi/-constraint method, are derived. The comparison reveals that for many problems, the simple algorithm SEMO is as efficient as this (1+1)-EA. For some problems, the improved variants FEMO and GEMO are provably better. For the analysis, we propose and apply two general tools, an upper bound technique based on a decision space partition and a randomized graph search algorithm, which facilitate the analysis considerably.     

Convergence analysis of some multiobjective evolutionary algorithms when discovering motifs

An important issue in multiobjective optimization is the study of the convergence speed of algorithms. An optimization problem must be defined as simple as possible to minimize the computational cost required to solve it. In this work, we study the convergence speed of seven multiobjective evolutionary algorithms: DEPT, MO-VNS, MOABC, MO-GSA, MO-FA, NSGA-II, and SPEA2; when solving an important biological problem: the motif discovery problem. We have used twelve instances of four different organisms as benchmark, analyzing the number of fitness function evaluations required by each algorithm to achieve reasonable quality solutions. We have used the hypervolume indicator to evaluate the solutions discovered by each algorithm, measuring its quality every 100 evaluations. This methodology also allows us to study the hit rates of the algorithms over 30 independent runs. Moreover, we have made a deeper study in the more complex instance of each organism. In this study, we observe the increase of the archive (number of non-dominated solutions) and the spread of the Pareto fronts obtained by the algorithm in the median execution. As we will see, our study reveals that DEPT, MOABC, and MO-FA provide the best convergence speeds and the highest hit rates.     

A parallel cooperative team of multiobjective evolutionary algorithms for motif discovery

When solving a wide range of complex scenarios of a given optimization problem, it is very difficult, if not impossible, to develop a single technique or algorithm that is able to solve all of them adequately. In this case, it is necessary to combine several algorithms by applying the most appropriate one in each case. Parallel computing can be used to improve the quality of the solutions obtained in a cooperative algorithms model. Exchanging information between parallel cooperative algorithms will alter their behavior in terms of solution searching, and it may be more effective than a sequential metaheuristic. For demonstrating this, a parallel cooperative team of four multiobjective evolutionary algorithms based on OpenMP is proposed for solving different scenarios of the Motif Discovery Problem (MDP), which is an important real-world problem in the biological domain. As we will see, the results show that the application of a properly configured parallel cooperative team achieves high quality solutions when solving the addressed problem, improving those achieved by the algorithms executed independently for a much longer time.     

一种新的分布性保持方法

分布性保持是多目标进化算法主要目标之一. 然而通常维护方法的性能与运行时间存在矛盾. 提出一种基于最小生成树的分布性维护方法. 利用最小生成树中的度数和边长对个体密度进行估计, 使低度数的边界个体和长边长的低密度个体得到了保留. 另外, 一次性选择个体进入下代种群, 避免了每移出一个个体就需要对个体密度进行调整的操作. 通过5个测试问题和4个方面的测试标准, 与3个著名的算法进行比较实验, 结果表明该方法在以较快速度对种群进行维护的同时, 拥有良好的分布性.     

Enhanced evolutionary algorithms for single and multiobjective optimization in the job shop scheduling problem

Over the past few years, a continually increasing number of research efforts have investigated the application of evolutionary computation techniques for the solution of scheduling problems. Scheduling can pose extremely complex combinatorial optimization problems, which belong to the NP-hard family. Last enhancements on evolutionary algorithms include new multirecombinative approaches. Multiple Crossovers Per Couple (MCPC) allows multiple crossovers on the couple selected for mating and Multiple Crossovers on Multiple Parents (MCMP) do this but on a set of more than two parents. Techniques for preventing incest also help to avoid premature convergence. Issues on representation and operators influence efficiency and efficacy of the algorithm. The present paper shows how enhanced evolutionary approaches, can solve the Job Shop Scheduling Problem (JSSP) in single and multiobjective optimization.     

Running time analysis of evolutionary algorithms on a simplified multiobjective knapsack problem

In this paper, the expected running time of two multiobjectiveevolutionary algorithms, SEMO and FEMO, is analyzed for a simpleinstance of the multiobjective 0/1 knapsack problem. The considered problem instance has two profit values per item andcannot be solved by one-bit mutations. In the analysis, we make use of two general upper bound techniques, thedecision space partition method and the graph search method. The paperdemonstrates how these methods, which have previously only beenapplied to algorithms with one-bit mutations, are equally applicablefor mutation operators where each bit is flipped independently with acertain probability.