Many-objective evolutionary algorithm based on adaptive weighted decomposition

Decomposition is a representative method for handling many-objective optimization problems with evolutionary algorithms. Classical decomposition scheme relies on a set of uniformly distributed reference vectors to divide the objective space into multiple subregions. This scheme often works poorly when the problem has an irregular Pareto front due to the inconsistency between the distribution of reference vectors and the shape of Pareto fronts. We propose in this paper an adaptive weighted decomposition based many-objective evolutionary algorithm to tackle complicated many-objective problems whose Pareto fronts may or may not be regular. Unlike traditional decomposition based algorithms that use a pre-defined set of reference vectors, the reference vectors in the proposed algorithm are produced from the population during the search. The experiments show that the performance of the proposed algorithm is competitive with other state-of-the-art algorithms and is less-sensitive to the irregularity of the Pareto fronts.     

Improved genetic algorithm optimization of water distribution system design by incorporating domain knowledge

Over the last two decades, evolutionary algorithms (EAs) have become a popular approach for solving water resources optimization problems. However, the issue of low computational efficiency limits their application to large, realistic problems. This paper uses the optimal design of water distribution systems (WDSs) as an example to illustrate how the efficiency of genetic algorithms (GAs) can be improved by using heuristic domain knowledge in the sampling of the initial population. A new heuristic procedure called the Prescreened Heuristic Sampling Method (PHSM) is proposed and tested on seven WDS cases studies of varying size. The EPANet input files for these case studies are provided as supplementary material. The performance of the PHSM is compared with that of another heuristic sampling method and two non-heuristic sampling methods. The results show that PHSM clearly performs best overall, both in terms of computational efficiency and the ability to find near-optimal solutions. In addition, the relative advantage of using the PHSM increases with network size.     

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。     

An efficient decomposition and dual-stage multi-objective optimization method for water distribution systems with multiple supply sources

This paper proposes an efficient decomposition and dual-stage multi-objective optimization (DDMO) method for designing water distribution systems with multiple supply sources (WDS-MSSs). Three phases are involved in the proposed DDMO approach. In Phase 1, an optimal source partitioning cut-set is identified for a WDS-MSS, allowing the entire WDS-MSS to be decomposed into sub-networks. Then in Phase 2 a non-dominated sorting genetic algorithm (NSGA-II) is employed to optimize the sub-networks separately, thereby producing an optimal front for each sub-network. Finally in Phase 3, another NSGA-II implementation is used to drive the combined sub-network front (an approximate optimal front) towards the Pareto front for the original complete WDS-MSS. Four WDS-MSSs are used to demonstrate the effectiveness of the proposed approach. Results obtained show that the proposed DDMO significantly outperforms the NSGA-II that optimizes the entire network as a whole in terms of efficiently finding good quality optimal fronts.     

基于幂变换的多目标进化算法MOEA?D权重设计方法

在多目标最优化问题中,如何求解一组均匀散布在前沿界面上的有效解具有重要意义.MOEA?D是最近出现的一种杰出的多目标进化算法,当前沿界面的形状是某种已知的类型时,MOEA?D使用高级分解的方法容易求出均匀散布在前沿界面上的有效解.然而,多目标优化问题的前沿界面的形状通常是未知的.为了使MOEA?D能求出一般多目标优化问题的均匀散布的有效解,利用幂函数对目标进行数学变换,使变换后的多目标优化问题的前沿界面在算法的进化过程中逐渐接近希望得到的形状,提出了一种求解一般的多目标优化问题的MOEA?D算法的权重设计方法,并且讨论了经过数学变换后前沿界面的保距性问题.采用建议的权重设计方法,MOEA?D更容易求出一般的多目标优化问题均匀散布的有效解.数值结果验证了算法的有效性.     

Simultaneous topology and sizing optimization of a water distribution network using a hybrid multiobjective evolutionary algorithm

This paper proposes a new direction for design optimization of a water distribution network (WDN). The new approach introduces an optimization process to the conceptual design stage of a WDN. The use of multiobjective evolutionary algorithms (MOEAs) for simultaneous topology and sizing design of piping networks is presented. The design problem includes both topological and sizing design variables while the objective functions are network cost and total head loss in pipes. The numerical technique, called a network repairing technique (NRT), is proposed to overcome difficulties in operating MOEAs for network topological design. The problem is then solved by using a number of established and newly developed MOEAs. Also, two new MOEAs namely multiobjective real code population-based incremental learning (RPBIL) and a hybrid algorithm of RPBIL with differential evolution (termed RPBIL–DE) are proposed to tackle the design problems. The optimum results obtained are illustrated and compared. It is shown that the proposed network repairing technique is an efficient and effective tool for topological design of WDNs. Based on the hypervolume indicator, the proposed RPBIL–DE is among the best MOEA performers.     

Multi-objective evolutionary optimization of polynomial neural networks for modelling and prediction of explosive cutting process

In this paper, evolutionary algorithms (EAs) are deployed for multi-objective Pareto optimal design of group method of data handling (GMDH)-type neural networks which have been used for modelling an explosive cutting process using some input–output experimental data. In this way, multi-objective EAs (non-dominated sorting genetic algorithm, NSGA-II) with a new diversity-preserving mechanism are used for Pareto optimization of such GMDH-type neural networks. The important conflicting objectives of GMDH-type neural networks that are considered in this work are, namely, training error (TE), prediction error (PE), and number of neurons (N) of such neural networks. Different pairs of theses objective functions are selected for 2-objective optimization processes. Therefore, optimal Pareto fronts of such models are obtained in each case which exhibit the trade-off between the corresponding pair of conflicting objectives and, thus, provide different non-dominated optimal choices of GMDH-type neural networks models for explosive cutting process. Moreover, all the three objectives are considered in a 3-objective optimization process, which consequently leads to some more non-dominated choices of GMDH-type models representing the trade-offs among the training error, prediction error, and number of neurons (complexity of network), simultaneously. The overlay graphs of these Pareto fronts also reveal that the 3-objective results include those of the 2-objective results and, thus, provide more optimal choices for the multi-objective design of GMDH-type neural networks in terms of minimum training error, minimum prediction error, and minimum complexity.     

Evolutionary algorithm with a directional local search for multiobjective optimization in combinatorial problems

Evolutionary algorithms (EAs) are often employed to multiobjective optimization, because they process an entire population of solutions which can be used as an approximation of the Pareto front of the tackled problem. It is a common practice to couple local search with evolutionary algorithms, especially in the context of combinatorial optimization. In this paper a new local search method is proposed that utilizes the knowledge concerning promising search directions. The proposed method can be used as a general framework and combined with many methods of iterating over a neighbourhood of an initial solution as well as various decomposition approaches. In the experiments the proposed local search method was used with an EA and tested on 2-, 3- and 4-objective versions of two well-known combinatorial optimization problems: the travelling salesman problem (TSP) and the quadratic assignment problem (QAP). For comparison two well-known local search methods, one based on Pareto dominance and the other based on decomposition, were used with the same EA. The results show that the EA coupled with the directional local search yields better results than the same EA coupled with any of the two reference methods on both the TSP and QAP problems.     

A Survey of Evolutionary Algorithms for Multi-Objective Optimization Problems With Irregular Pareto Fronts

Evolutionary algorithms have been shown to be very successful in solving multi-objective optimization problems(MOPs).However,their performance often deteriorates when solving MOPs with irregular Pareto fronts.To remedy this issue,a large body of research has been performed in recent years and many new algorithms have been proposed.This paper provides a comprehensive survey of the research on MOPs with irregular Pareto fronts.We start with a brief introduction to the basic concepts,followed by a summary of the benchmark test problems with irregular problems,an analysis of the causes of the irregularity,and real-world optimization problems with irregular Pareto fronts.Then,a taxonomy of the existing methodologies for handling irregular problems is given and representative algorithms are reviewed with a discussion of their strengths and weaknesses.Finally,open challenges are pointed out and a few promising future directions are suggested.     

Utopian point based decomposition for multi-objective optimization problems with complicated Pareto fronts

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) has been considered as a promising method for solving multi-objective optimization problems (MOPs). It devotes most of its effort on convergence by optimizing a set of scalar optimization subproblems in a collaborative manner, while maintaining the diversity by using a set of uniformly distributed weight vectors. However, more recent studies illustrated that MOEA/D faces difficulties on MOPs with complicated Pareto fronts, mainly because the uniformity of weight vectors no longer lead to an evenly scattered approximation of the Pareto fronts in these cases. To remedy this, we suggest replacing the ideal point in the reciprocal Tchebycheff decomposition method with a more optimistic utopian point, with the aim of alleviating the sensitivity of MOEA/D to the Pareto front shape of MOPs. Experimental studies on benchmark and real-world problems have shown that such simple modification can significantly improve the performances of MOEA/D with reciprocal Tchebycheff decomposition on MOPs with complicated Pareto fronts.     

A hierarchical optimization neural network for large-scale dynamic systems

A recurrent neural network for dynamical hierarchical optimization of nonlinear discrete large-scale systems is presented. The proposed neural network consists of hierarchically structured sub-networks: one coordination sub-network at the upper level and several local optimization sub-networks at the lower level. In particular, the coordination sub-network and the local optimization sub-networks work simultaneously. This feature makes the proposed method outperform in computational efficiency the conventional iterative algorithms where there usually exists an alternately waiting time during the coordination and local optimization processes. Moreover, the state equations of the subsystems of the large-scale system are imbedded into their corresponding local optimization sub-networks. This imbedding technique not only overcomes the difficulty in treating the constraints imposed by the state equations, but also leads to significant reduction in the network size. We present stability analysis to prove that the neural network is asymptotically stable and this stable state corresponds to the optimal solution to the original optimal control problem. Finally, we illustrate the performance of the proposed method by an example.     

An analysis of the interface between evolutionary algorithm operators and problem features for water resources problems. A case study in water distribution network design

Evolutionary Algorithms (EAs) have been widely employed to solve water resources problems for nearly two decades with much success. However, recent research in hyperheuristics has raised the possibility of developing optimisers that adapt to the characteristics of the problem being solved. In order to select appropriate operators for such optimisers it is necessary to first understand the interaction between operator and problem. This paper explores the concept of EA operator behaviour in real world applications through the empirical study of performance using water distribution networks (WDN) as a case study. Artificial networks are created to embody specific WDN features which are then used to evaluate the impact of network features on operator performance. The method extracts key attributes of the problem which are encapsulated in the natural features of a WDN, such as topologies and assets, on which different EA operators can be tested. The method is demonstrated using small exemplar networks designed specifically so that they isolate individual features. A set of operators are tested on these artificial networks and their behaviour characterised. This process provides a systematic and quantitative approach to establishing detailed information about an algorithm's suitability to optimise certain types of problem. The experiment is then repeated on real-world inspired networks and the results are shown to fit with the expected results.     

Multiobjective evolutionary algorithms: a comparative case studyand the strength Pareto approach

Evolutionary algorithms (EAs) are often well-suited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in a single run. However, the few comparative studies of different methods presented up to now remain mostly qualitative and are often restricted to a few approaches. In this paper, four multiobjective EAs are compared quantitatively where an extended 0/1 knapsack problem is taken as a basis. Furthermore, we introduce a new evolutionary approach to multicriteria optimization, the strength Pareto EA (SPEA), that combines several features of previous multiobjective EAs in a unique manner. It is characterized by (a) storing nondominated solutions externally in a second, continuously updated population, (b) evaluating an individual's fitness dependent on the number of external nondominated points that dominate it, (c) preserving population diversity using the Pareto dominance relationship, and (d) incorporating a clustering procedure in order to reduce the nondominated set without destroying its characteristics. The proof-of-principle results obtained on two artificial problems as well as a larger problem, the synthesis of a digital hardware-software multiprocessor system, suggest that SPEA can be very effective in sampling from along the entire Pareto-optimal front and distributing the generated solutions over the tradeoff surface. Moreover, SPEA clearly outperforms the other four multiobjective EAs on the 0/1 knapsack problem     

基于目标迁移和条件替代的高维多目标进化算法

尽管许多高维多目标进化算法已被提出,但大多仍无法有效处理具有不规则Pareto前沿的高维多目标优化问题.鉴于此,提出基于目标迁移和条件替代的高维多目标进化算法(MaOEA-OTCR),在环境选择过程中利用目标迁移策略和条件替代准则协作逐一选择收敛性和多样性好的个体进入下一代.前者首先选择位于Pareto前沿边界的极值解进入下一代,以确定Pareto前沿的范围,同时选择收敛性最好的若干个体进入下一代,以加速种群收敛;然后迁移已选解集且利用迁移解集和未迁移解集的最大距离来选择收敛性和多样性好的个体进入下一代.后者利用基于角度和收敛性评估的条件取代准则来防止前者过度强调多样性.此外,提出一个多标准决策的匹配选择策略,旨在增加具有良好收敛性和多样性种群个体结合的概率,进一步提升算法的搜索效率.为了验证MaOEA-OTCR的有效性,在3个测试集上与8个先进的高维多目标进化算法进行对比实验.实验结果表明, MaOEA-OTCR在处理高维多目标优化问题时不仅能够获得较强的竞争性能,而且有能力处理具有不规则Pareto前沿的高维多目标优化问题.     

参考点自适应调整下评价指标驱动的高维多目标进化算法

在具有不同Pareto前沿形状的优化问题上, 基于参考点的高维多目标进化算法表现出较差的通用性. 为了解决这个问题, 提出参考点自适应调整下评价指标驱动的高维多目标进化算法(Many-objective evolutionary algorithm driven by evaluation indicator under adaptive reference point adjustment, MaOEA-IAR). MaOEA-IAR提出Pareto前沿形状监测基础上的参考点自适应策略, 利用该策略选择一组候选解作为初始参考点; 然后通过曲线参数对参考点位置进行调整; 将最终得到的能够适应不同Pareto前沿的参考点用于计算增强的反世代距离指标, 基于指标值设计适应度函数作为选择标准. 实验证明提出的算法在处理各种Pareto前沿形状的优化问题时能获得较好的性能, 算法通用性高.     

C-PSA: Constrained Pareto simulated annealing for constrained multi-objective optimization

In recent years, evolutionary algorithms (EAs) have been extensively developed and utilized to solve multi-objective optimization problems. However, some previous studies have shown that for certain problems, an approach which allows for non-greedy or uphill moves (unlike EAs), can be more beneficial. One such approach is simulated annealing (SA). SA is a proven heuristic for solving numerical optimization problems. But owing to its point-to-point nature of search, limited efforts has been made to explore its potential for solving multi-objective problems. The focus of the presented work is to develop a simulated annealing algorithm for constrained multi-objective problems. The performance of the proposed algorithm is reported on a number of difficult constrained benchmark problems. A comparison with other established multi-objective optimization algorithms, such as infeasibility driven evolutionary algorithm (IDEA), Non-dominated sorting genetic algorithm II (NSGA-II) and multi-objective Scatter search II (MOSS-II) has been included to highlight the benefits of the proposed approach.     

An efficient multiobjective differential evolution algorithm for engineering design

Solving engineering design and resources optimization via multiobjective evolutionary algorithms (MOEAs) has attracted much attention in the last few years. In this paper, an efficient multiobjective differential evolution algorithm is presented for engineering design. Our proposed approach adopts the orthogonal design method with quantization technique to generate the initial archive and evolutionary population. An archive (or secondary population) is employed to keep the nondominated solutions found and it is updated by a new relaxed form of Pareto dominance, called Pareto-adaptive ϵ-dominance (paϵ-dominance), at each generation. In addition, in order to guarantee to be the best performance produced, we propose a new hybrid selection mechanism to allow the archive solutions to take part in the generating process. To handle the constraints, a new constraint-handling method is employed, which does not need any parameters to be tuned for constraint handling. The proposed approach is tested on seven benchmark constrained problems to illustrate the capabilities of the algorithm in handling mathematically complex problems. Furthermore, four well-studied engineering design optimization problems are solved to illustrate the efficiency and applicability of the algorithm for multiobjective design optimization. Compared with Nondominated Sorting Genetic Algorithm II, one of the best MOEAs available at present, the results demonstrate that our approach is found to be statistically competitive. Moreover, the proposed approach is very efficient and is capable of yielding a wide spread of solutions with good coverage and convergence to true Pareto-optimal fronts.     

多目标协调进化算法研究

进化算法适合解决多目标优化问题,但难以产生高维优化问题的最优解,文中针对此问题提出了一种求解高维目标优化问题的新进化方法,即多目标协调进化算法,主要特点是进化群体按协调模型使用偏好信息进行偏好排序,而不是基于Pareto优于关系进行了个体排序,实验结果表明,所提出的算法是可行而有效的,且能在有限进化代数内收敛。     

Multiobjective optimization using an adaptive weighting scheme

A new Pareto front approximation method is proposed for multiobjective optimization problems (MOPs) with bound constraints. The method employs a hybrid optimization approach using two derivative-free direct search techniques, and intends to solve black box simulation-based MOPs where the analytical form of the objectives is not known and/or the evaluation of the objective function(s) is very expensive. A new adaptive weighting scheme is proposed to convert a multiobjective optimization problem to a single objective optimization problem. Another contribution of this paper is the generalization of the star discrepancy-based performance measure for problems with more than two objectives. The method is evaluated using five test problems from the literature, and a realistic engineering problem. Results show that the method achieves an arbitrarily close approximation to the Pareto front with a good collection of well-distributed nondominated points for all six test problems.     

POD-based error control for reduced-order bicriterial PDE-constrained optimization

In the present paper, a bicriterial optimal control problem governed by an abstract evolution problem and bilateral control constraints is considered. To compute Pareto optimal points and the Pareto front numerically, the (Euclidean) reference point method is applied, where many scalar constrained optimization problems have to be solved. For this reason a reduced-order approach based on proper orthogonal decomposition (POD) is utilized. An a-posteriori error analysis ensures a desired accuracy for the Pareto optimal points and for the Pareto front computed by the POD method. Numerical experiments for evolution problems with convection-diffusion illustrate the efficiency of the presented approach.