可求解多目标优化问题的演化博弈优化算法

借鉴演化博弈的思想和选择机制,提出了一种新的基于演化博弈的优化算法(EGOA)用于多目标问题的求解.算法框架具备对该类问题的通用性.为了对算法性能进行评估,采用了一组多目标优化问题(MOPs)的测试函数进行实验.实验结果表明,使用本算法搜索得到的演化稳定策略集合能够很好地逼近多目标优化问题的帕累托前沿,与一些经典的演化算法相比具有良好的问题求解能力.     

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.     

The Nash Equilibrium Point of Dynamic Games Using Evolutionary Algorithms in Linear Dynamics and Quadratic System

In this paper, examining some games, we show that classical techniques are not always effective for games with not many stages and players and it can’t be claimed that these techniques of solution always obtain the optimal and actual Nash equilibrium point. For solving these problems, two evolutionary algorithms are then presented based on the population to solve general dynamic games. The first algorithm is based on the genetic algorithm and we use genetic algorithms to model the players' learning process in several models and evaluate them in terms of their convergence to the Nash Equilibrium. in the second algorithm, a Particle Swarm Intelligence Optimization (PSO) technique is presented to accelerate solutions’ convergence. It is claimed that both techniques can find the actual Nash equilibrium point of the game keeping the problem’s generality and without imposing any limitation on it and without being caught by the local Nash equilibrium point. The results clearly show the benefits of the proposed approach in terms of both the quality of solutions and efficiency.     

Game model-based co-evolutionary algorithm with non-dominated memory and Euclidean distance selection mechanisms for multi-objective optimization

Many real-world problems involve simultaneous optimization of several incommensurable and often competing objectives. In the search for solutions to multi-objective optimization problems (MOPs), we find that there is no single optimum but rather a set of optimums known as the “Pareto optimal set”. Co-evolutionary algorithms are well suited to optimization problems which involve several often competing objectives. Co-evolutionary algorithms are aimed at evolving individuals through individuals competing in an objective space. In order to approximate the ideal Pareto optimal set, the search capability of diverse individuals in an objective space can be used to determine the performance of evolutionary algorithms. Non-dominated memory and Euclidean distance selection mechanisms for co-evolutionary algorithms have the goal of overcoming the limited search capability of diverse individuals in the population space. In this paper, we propose a method for maintaining population diversity in game model-based co-evolutionary algorithms, and we evaluate the effectiveness of our approach by comparing it with other methods through rigorous experiments on several MOPs.     

Coevolutionary augmented Lagrangian methods for constrainedoptimization

This paper introduces a coevolutionary method developed for solving constrained optimization problems. This algorithm is based on the evolution of two populations with opposite objectives to solve saddle-point problems. The augmented Lagrangian approach is taken to transform a constrained optimization problem to a zero-sum game with the saddle point solution. The populations of the parameter vector and the multiplier vector approximate the zero-sum game by a static matrix game, in which the fitness of individuals is determined according to the security strategy of each population group. Selection, recombination, and mutation are done by using the evolutionary mechanism of conventional evolutionary algorithms such as evolution strategies, evolutionary programming, and genetic algorithms. Four benchmark problems are solved to demonstrate that the proposed coevolutionary method provides consistent solutions with better numerical accuracy than other evolutionary methods     

一种基于输运理论的多目标演化算法

提出了一种根据输运理论中的粒子输运方程、相空间能量定律和熵增法则构造的一种能够准确、高效地求解多目标优化问题的多目标演化算法(MOPEA).由于该算法使用了粒子系统从非平衡达到平衡的理论来定义求解多目标问题的Rank函数和Niche适应值函数,使得种群中的所有个体都有机会参与演化操作,以达到快速、均匀地求出多目标优化问题的Pareto最优解.数据实验显示,利用该算法求解多目标优化问题不仅能够使算法快速地收敛到全局Pareto前沿,同时由于该算法要求所有的粒子都要参与杂交和变异等演化操作,从而避免问题早熟现象的出现,并通过与传统演化算法的性能指标分析比较说明,使用该算法求解多目标优化问题具有明显的优越性.     

基于改进量子粒子群算法的Nash均衡研究

在N人非合作博弈Nash均衡问题求解过程中,将量子不确定性原理、协同演化以及免疫算法内的抗体浓度抑制机制引进到经典粒子群算法中,设计了一种新型改进量子粒子群算法来更好地处理Nash均衡问题。该算法在运算过程中,运用抗体浓度以及协同演化的方式来维系粒子群具备的多样性特征,并借助量子不确定性缩减迭代搜索耗时。该算法不仅有效地将粒子群算法运算简单与方便实现的特质承继下来,而且算法的收敛速度以及其全局搜索能力都获得了大幅度的提升。相关数值算例分析表明,改进的算法能够更好地处理粒子早熟,相较遗传算法以及免疫粒子群算法更具性能优越性。     

基于Pareto熵的多目标粒子群优化算法

粒子群优化算法因形式简洁、收敛快速和参数调节机制灵活等优点,同时一次运行可得到多个解,且能逼近非凸或不连续的Pareto最优前端,因而被认为是求解多目标优化问题最具潜力的方法之一.但当粒子群优化算法从单目标问题扩展到多目标问题时,Pareto最优解集的存储与维护、全局和个体最优解的选择以及开发与开采的平衡等问题亦随之出现.通过目标空间变换方法,采用Pareto前端在被称为平行格坐标系统的新目标空间中的分布熵及差熵评估种群的多样性及进化状态,并以此为反馈信息来设计进化策略,使得算法能够兼顾近似Pareto前端的收敛性和多样性.同时,引入格占优和格距离密度的概念来评估Pareto最优解的个体环境适应度,以此建立外部档案更新方法和全局最优解选择机制,最终形成了基于Pareto熵的多目标粒子群优化算法.实验结果表明：在IGD性能指标上,与另外8种对等算法相比,该算法在由ZDT和DTLZ系列组成的12个多目标测试问题集中表现出了显著的性能优势.     

Data Structures in Multi-Objective Evolutionary Algorithms

Data structures used for an algorithm can have a great impact on its performance,particularly for the solution of large and complex problems,such as multi-objective optimization problems(MOPs).Multi-objective evolutionary algorithms(MOEAs) are considered an attractive approach for solving MOPs,since they are able to explore several parts of the Pareto front simultaneously.The data structures for storing and updating populations and non-dominated solutions(archives) may affect the efficiency of the search process.This article describes data structures used in MOEAs for realizing populations and archives in a comparative way,emphasizing their computational requirements and general applicability reported in the original work.     

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.     

一种基于混合高斯模型的多目标进化算法

目前,大多数多目标进化算法采用为单目标优化所设计的重组算子.通过证明或实验分析了几个典型的单目标优化重组算子并不适合某些多目标优化问题.提出了基于分解技术和混合高斯模型的多目标优化算法（multiobjective evolutionary algorithm based on decomposition and mixture Gaussian models,简称MOEA/D-MG）.该算法首先采用一个改进的混合高斯模型对群体建模并采样产生新个体,然后利用一个贪婪策略来更新群体.针对具有复杂Pareto前沿的多目标优化问题的测试结果表明,对给定的大多数测试题,该算法具有良好的效果.     

Hybrid coevolutionary programming for Nash equilibrium search in games with local optima

The conventional local optimization path and coevolutionary processes are studied when "local Nash equilibrium (NE) traps" exist. Conventional NE search algorithms in games with local optima can misidentify NE by following a local optimization path. We prove that any iterative NE search algorithms based on local optimization cannot differentiate real NE and "local NE traps". Coevolutionary programming, a parallel and global search algorithm, is applied to overcome this problem. In order to enhance the poor convergence of simple coevolutionary programming, hybrid coevolutionary programming is suggested. The conventional NE algorithms, simple coevolutionary programming, and hybrid coevolutionary algorithms are tested through a simple numerical example and transmission-constrained electricity market examples.     

Multi-objective differential evolution with performance-metric-based self-adaptive mutation operator for chemical and biochemical dynamic optimization problems

Each mutation operator of differential evolution (DE) algorithm is generally suitable for certain specific types of multi-objective optimization problems (MOPs) or particular stages of the evolution. To automatically select an appropriate mutation operator for solving MOPs in different phases of the evolution, a multi-objective differential evolution with performance-metric-based self-adaptive mutation operator (MODE-PMSMO) is proposed in this study. In MODE-PMSMO, a modified inverted generational distance (IGD) is utilized to evaluate the performance of each mutation operator and guide the evolution of mutation operators. The proposed MODE-PMSMO is then compared with seven multi-objective evolutionary algorithms (MOEAs) on five bi-objective and five tri-objective optimization problems. Generally, MODE-PMSMO exhibits the best average performance among all compared algorithms on ten MOPs. Additionally, MODE-PMSMO is employed to solve four typical multi-objective dynamic optimization problems in chemical and biochemical processes. Experimental results indicate that MODE-PMSMO is suitable for solving these actual problems and can provide a set of nondominated solutions for references of decision makers.     

双精英协同进化遗传算法

针对传统遗传算法早熟收敛和收敛速度慢的问题,提出一种双精英协同进化遗传算法(double elite coevolutionary genetic algorithm,简称DECGA).该算法借鉴了精英策略和协同进化的思想,选择两个相异的、高适应度的个体(精英个体)作为进化操作的核心,两个精英个体分别按照不同的评价函数来选择个体,组成各自的进化子种群.两个子种群分别采用不同的进化策略,以平衡算法的勘探和搜索能力.理论分析证明,该算法具有全局收敛性.通过对测试函数的实验,其结果表明,该算法能搜索到几乎所有测试函数的最优解,同时能够有效地保持种群的多样性.与已有算法相比,该算法在收敛速度和搜索全局最优解上都有了较大的改进和提高.     

A game-theoretic and dynamical-systems analysis of selection methods in coevolution

We use evolutionary game theory (EGT) to investigate the dynamics and equilibria of selection methods in coevolutionary algorithms. The canonical selection method used in EGT is equivalent to the standard "fitness-proportional" selection method used in evolutionary algorithms. All attractors of the EGT dynamic are Nash equilibria; we focus on simple symmetric variable-sum games that have polymorphic Nash-equilibrium attractors. Against the dynamics of proportional selection, we contrast the behaviors of truncation selection, (/spl mu/,/spl lambda/),(/spl mu/+/spl lambda/), linear ranking, Boltzmann, and tournament selection. Except for Boltzmann selection, each of the methods we test unconditionally fail to achieve polymorphic Nash equilibrium. Instead, we find point attractors that lack game-theoretic justification, cyclic dynamics, or chaos. Boltzmann selection converges onto polymorphic Nash equilibrium only when selection pressure is sufficiently low; otherwise, we obtain attracting limit-cycles or chaos. Coevolutionary algorithms are often used to search for solutions (e.g., Nash equilibria) of games of strategy; our results show that many selection methods are inappropriate for finding polymorphic Nash solutions to variable-sum games. Another application of coevolution is to model other systems; our results emphasize the degree to which the model's behavior is sensitive to implementation details regarding selection-details that we might not otherwise believe to be critical.     

基于聚类的快速多目标遗传算法

多目标遗传算法非常适合于求解多目标优化问题．讨论了进化个体之间的支配关系及有关性质，论证了可以用快速排序的方法对进化群体中的个体进行分类，同时探讨了用聚类方法来保持群体的多样性，具体讨论了基于层次凝聚距离的聚类，在此基础上提出了用分类和聚类的方法构造新的进化群体．理论分析与实验结果表明，所讨论的方法比较国际上已有的方法具有更快的收敛速度．     

Multi-objective modified differential evolution algorithm with archive-base mutation for solving multi-objective $$$$-xylene oxidation rocess

Maximizing the diversity of the obtained objective vectors and increasing the convergence speed to the true Pareto front are two important issues in the design of multi-objective evolutionary algorithms (MOEAs). To solve complex multi-objective optimization problems (MOPs), a multi-objective modified differential evolution algorithm with archive-base mutation (MOMDE-AM) is proposed. In MOMDE-AM, with the purpose of reducing the loss of population evolution information, a modified mutation strategy with archive is introduced, which could utilize several useful inferior solutions and provide promising direction information toward the true Pareto front. The performance of MOMDE-AM is compared with five other MOEAs on five bi-objective and five tri-objective optimization problems. The simulation and statistical analysis results indicate that the overall performance of MOMDE-AM is better than those of the compared algorithms on these test functions. Finally, MOMDE-AM is used to optimize ten operation conditions of the \(p\)-xylene oxidation reaction process; the results show that MOMDE-AM is an effective and efficient optimization tool for solving actual MOPs.     

Self-adaptive Bat Algorithm With Genetic Operations

Swarm intelligence in a bat algorithm (BA) provides social learning. Genetic operations for reproducing individuals in a genetic algorithm (GA) offer global search ability in solving complex optimization problems. Their integration provides an opportunity for improved search performance. However, existing studies adopt only one genetic operation of GA, or design hybrid algorithms that divide the overall population into multiple subpopulations that evolve in parallel with limited interactions only. Differing from them, this work proposes an improved self-adaptive bat algorithm with genetic operations (SBAGO) where GA and BA are combined in a highly integrated way. Specifically, SBAGO performs their genetic operations of GA on previous search information of BA solutions to produce new exemplars that are of high-diversity and high-quality. Guided by these exemplars, SBAGO improves both BA’s efficiency and global search capability. We evaluate this approach by using 29 widely-adopted problems from four test suites. SBAGO is also evaluated by a real-life optimization problem in mobile edge computing systems. Experimental results show that SBAGO outperforms its widely-used and recently proposed peers in terms of effectiveness, search accuracy, local optima avoidance, and robustness.      

一种多目标优化问题的理想灰色粒子群算法*

针对逼近理想解的排序方法对Pareto前端的距离跟踪以及灰色关联度能够很好地分析非劣解集曲线与Pareto最优解集曲线的相似性,提出了一种求解多目标优化问题的理想灰色粒子群算法。该算法利用理想解理论与灰色关联度理论来求解粒子与理想解之间的相对适应度和灰色关联度系数,把两者的和定义为相对理想度,通过相对理想度来判别粒子的优劣,以确定个体极值和全局极值。通过四组不同类型的基准函数测试算法性能,并与目标加权法和灰色粒子群算法比较分析,结果表明该算法能够较好地收敛到Pareto最优解集,不但具有较好的收敛性和分布     

Convergence of multi-objective evolutionary algorithms to a uniformly distributed representation of the Pareto front

In evolutionary multi-objective optimization (EMO), the convergence to the Pareto set of a multi-objective optimization problem (MOP) and the diversity of the final approximation of the Pareto front are two important issues. In the existing definitions and analyses of convergence in multi-objective evolutionary algorithms (MOEAs), convergence with probability is easily obtained because diversity is not considered. However, diversity cannot be guaranteed. By combining the convergence with diversity, this paper presents a new definition for the finite representation of a Pareto set, the B-Pareto set, and a convergence metric for MOEAs. Based on a new archive-updating strategy, the convergence of one such MOEA to the B-Pareto sets of MOPs is proved. Numerical results show that the obtained B-Pareto front is uniformly distributed along the Pareto front when, according to the new definition of convergence, the algorithm is convergent.