Solution of multiobjective optimization problems: coevolutionary algorithm based on evolutionary game theory

When attempting to solve multiobjective optimization problems (MOPs) using evolutionary algorithms, the Pareto genetic algorithm (GA) has now become a standard of sorts. After its introduction, this approach was further developed and led to many applications. All of these approaches are based on Pareto ranking and use the fitness sharing function to keep diversity. On the other hand, the scheme for solving MOPs presented by Nash introduced the notion of Nash equilibrium and aimed at solving MOPs that originated from evolutionary game theory and economics. Since the concept of Nash Equilibrium was introduced, game theorists have attempted to formalize aspects of the evolutionary equilibrium. Nash genetic algorithm (Nash GA) is the idea to bring together genetic algorithms and Nash strategy. The aim of this algorithm is to find the Nash equilibrium through the genetic process. Another central achievement of evolutionary game theory is the introduction of a method by which agents can play optimal strategies in the absence of rationality. Through the process of Darwinian selection, a population of agents can evolve to an evolutionary stable strategy (ESS). In this article, we find the ESS as a solution of MOPs using a coevolutionary algorithm based on evolutionary game theory. By applying newly designed coevolutionary algorithms to several MOPs, we can confirm that evolutionary game theory can be embodied by the coevolutionary algorithm and this coevolutionary algorithm can find optimal equilibrium points as solutions for an MOP. We also show the optimization performance of the co-evolutionary algorithm based on evolutionary game theory by applying this model to several MOPs and comparing the solutions with those of previous evolutionary optimization models. This work was presented, in part, at the 8th International Symposium on Artificial Life and Robotics, Oita, Japan, January 24#x2013;26, 2003.     

A Tournament-Based Competitive Coevolutionary Algorithm

For an efficient competitive coevolutionary algorithm, it is important that competing populations be capable of maintaining a coevolutionary balance and hence, continuing evolutionary arms race to increase the levels of complexity. We propose a competitive coevolutionary algorithm that combines the strategies of neighborhood-based evolution, entry fee exchange tournament competition (EFE-TC) and localized elitism. An emphasis is placed on analyzing the effects of these strategies on the performance of competitive coevolutionary algorithms. We have tested the proposed algorithm with two adversarial problems: sorting network and Nim game problems that have different characteristics. The experimental results show that the interacting effects of the strategies appear to promote a balanced evolution between host and parasite populations, which naturally leads them to keep on evolutionary arms race. Consequently, the proposed algorithm provides good quality solutions with a little computation time.     

Self-adaptive differential evolution algorithm with α-constrained-domination principle for constrained multi-objective optimization

Real-world problems are inherently constrained optimization problems often with multiple conflicting objectives. To solve such constrained multi-objective problems effectively, in this paper, we put forward a new approach which integrates self-adaptive differential evolution algorithm with α-constrained-domination principle, named SADE-αCD. In SADE-αCD, the trial vector generation strategies and the DE parameters are gradually self-adjusted adaptively based on the knowledge learnt from the previous searches in generating improved solutions. Furthermore, by incorporating domination principle into α-constrained method, α-constrained-domination principle is proposed to handle constraints in multi-objective problems. The advantageous performance of SADE-αCD is validated by comparisons with non-dominated sorting genetic algorithm-II, a representative of state-of-the-art in multi-objective evolutionary algorithms, and constrained multi-objective differential evolution, over fourteen test problems and four well-known constrained multi-objective engineering design problems. The performance indicators show that SADE-αCD is an effective approach to solving constrained multi-objective problems, which is basically enabled by the integration of self-adaptive strategies and α-constrained-domination principle.     

A differential evolution approach for solving constrained min–max optimization problems

Many problems occurring in engineering can be formulated as min–max optimization problems, for instance, in game theory, robust optimal control and many others. Min–max problems are considered difficult to solve, specially constrained min–max optimization problems. Approaches using co-evolutionary algorithms have successfully been used to solve min–max optimization problems without constraints. We propose a novel differential evolution approach consisting of three populations with a scheme of copying individuals for solving constrained min–max problems. Promising results have been obtained showing the suitability of the approach.     

An Endosymbiotic Evolutionary Algorithm for Optimization

This paper proposes a new symbiotic evolutionary algorithm to solve complex optimization problems. This algorithm imitates the natural evolution process of endosymbionts, which is called endosymbiotic evolutionary algorithm. Existing symbiotic algorithms take the strategy that the evolution of symbionts is separated from the host. In the natural world, prokaryotic cells that are originally independent organisms are combined into an eukaryotic cell. The basic idea of the proposed algorithm is the incorporation of the evolution of the eukaryotic cells into the existing symbiotic algorithms. In the proposed algorithm, the formation and evolution of the endosymbionts is based on fitness, as it can increase the adaptability of the individuals and the search efficiency. In addition, a localized coevolutionary strategy is employed to maintain the population diversity. Experimental results demonstrate that the proposed algorithm is a promising approach to solving complex problems that are composed of multiple sub- problems interrelated with each other.     

Evolutionary Algorithms for Minimax Problems in Robust Design

Many robust design problems can be described by minimax optimization problems. Classical techniques for solving these problems have typically been limited to a discrete form of the problem. More recently, evolutionary algorithms, particularly coevolutionary optimization techniques, have been applied to minimax problems. A new method of solving minimax optimization problems using evolutionary algorithms is proposed. The performance of this algorithm is shown to compare favorably with the existing methods on test problems. The performance of the algorithm is demonstrated on a robust pole placement problem and a ship engineering plant design problem.     

ε constrained differential evolution with pre-estimated comparison using gradient-based approximation for constrained optimization problems

Many real-world problems can be categorized as constrained optimization problems. So, designing effective algorithms for constrained optimization problems become more and more important. In designing algorithms, how to guide the individuals moving more efficiently towards the feasible region is one of the most important aspects on finding the optimum of constrained optimization problems. In this paper, we propose an improved ε constrained differential evolution, which combines with pre-estimated comparison gradient based approximation. The proposed algorithm uses gradient matrix to determine whether the trail vector generated by differential evolution algorithm is worth using the fitness function to evaluate it or not. Pre-estimated comparison gradient based approximation is used as a detector to find the promising offspring and in this way can we guide the individuals moving towards the feasible region. The proposed method is tested both on twenty-four benchmark functions and four well-known engineering optimization problems. Experimental results show that the proposed algorithm is highly competitive in comparing with other state-of-the-art algorithms. The proposed algorithm offers higher accuracy in engineering optimization problems for constrained optimization problems.     

Target shape design optimization by evolving B-splines with cooperative coevolution

With high reputation in handling non-linear and multi-model problems with little prior knowledge, evolutionary algorithms (EAs) have successfully been applied to design optimization problems as robust optimizers. Since real-world design optimization is often computationally expensive, target shape design optimization problems (TSDOPs) have been frequently used as efficient miniature model to check algorithmic performance for general shape design. There are at least three important issues in developing EAs for TSDOPs, i.e., design representation, fitness evaluation and evolution paradigm. Existing work has mainly focused on the first two issues, in which (1) an adaptive encoding scheme with B-spline has been proposed as a representation, and (2) a symmetric Hausdorff distance based metric has been used as a fitness function. But for the third issue, off-the-shelf EAs were used directly to evolve B-spline control points and/or knot vector. In this paper, we first demonstrate why it is unreasonable to evolve the control points and knot vector simultaneously. And then a new coevolutionary paradigm is proposed to evolve the control points and knot vector of B-spline separately in a cooperative manner. In the new paradigm, an initial population is generated for both the control points, and the knot vector. The two populations are evolved mostly separately in a round-robin fashion, with only cooperation at the fitness evaluation phase. The new paradigm has at least two significant advantages over conventional EAs. Firstly, it provides a platform to evolve both the control points and knot vector reasonably. Secondly, it reduces the difficulty of TSDOPs by decomposing the objective vector into two smaller subcomponents (i.e., control points and knot vector). To evaluate the efficacy of the proposed coevolutionary paradigm, an algorithm named CMA-ES-CC was formulated. Experimental studies were conducted based on two target shapes. The comparison with six other EAs suggests that the proposed cooperative coevolution paradigm is very effective for TSDOPs.     

协同演化算法研究进展

协同演化算法(coevolutionary algorithms,CEA)是当前国际上计算智能研究的一个热点,它运用生物协同演化的思想,是针对演化算法的不足而兴起的,通过构造两个或多个种群,建立它们之间的竞争或合作关系,多个种群通过相互作用来提高各自性能,适应复杂系统的动态演化环境,以达到种群优化的目的.介绍了协同演化算法的研究状况以及目前的研究进展,概述了它的基本算法、主要特点、理论与技术,同时介绍了一些主要的应用领域,指出了协同演化算法的研究方向.     

Modeling and convergence analysis of distributed coevolutionary algorithms

A theoretical foundation is presented for modeling and convergence analysis of a class of distributed coevolutionary algorithms applied to optimization problems in which the variables are partitioned among p nodes. An evolutionary algorithm at each of the p nodes performs a local evolutionary search based on its own set of primary variables, and the secondary variable set at each node is clamped during this phase. An infrequent intercommunication between the nodes updates the secondary variables at each node. The local search and intercommunication phases alternate, resulting in a cooperative search by the p nodes. First, we specify a theoretical basis for a class of centralized evolutionary algorithms in terms of construction and evolution of sampling distributions over the feasible space. Next, this foundation is extended to develop a model for a class of distributed coevolutionary algorithms. Convergence and convergence rate analyzes are pursued for basic classes of objective functions. Our theoretical investigation reveals that for certain unimodal and multimodal objectives, we can expect these algorithms to converge at a geometrical rate. The distributed coevolutionary algorithms are of most interest from the perspective of their performance advantage compared to centralized algorithms, when they execute in a network environment with significant local access and internode communication delays. The relative performance of these algorithms is therefore evaluated in a distributed environment with realistic parameters of network behavior.     

A multi-population cultural algorithm with adaptive diversity preservation and its application in ammonia synthesis process

A multi-population cultural differential evolution (MCDE) algorithm is proposed. Each of the populations is managed by its private cultural differential evolution algorithm, in which a center individual is introduced into the belief space and selection function follows a new method to select the offspring for the next generation. To accelerate the convergence speed, the populations exchange their knowledge with each other every given generations. An adaptive mechanism of population diversity preservation is put forward to prevent the populations from being trapped in local optima. In the adaptive mechanism, the idea of culture fusion between populations is used to know the convergence status, so that the diversity of populations is kept along the evolutionary process. The performance evaluation on MCDE using eleven constrained optimization problems shows that MCDE is a competitive approach. MCDE is further applied to a practical optimization problem in an ammonia synthesis system with the objective to maximize the net value of ammonia. The results achieved by MCDE are compared with those by two traditional differential evolution algorithms, which indicate that MCDE has more excellent performance and better effectiveness.     

A multi-objective differential evolutionary algorithm for constrained multi-objective optimization problems with low feasible ratio

Most current evolutionary multi-objective optimization (EMO) algorithms perform well on multi-objective optimization problems without constraints, but they encounter difficulties in their ability for constrained multi-objective optimization problems (CMOPs) with low feasible ratio. To tackle this problem, this paper proposes a multi-objective differential evolutionary algorithm named MODE-SaE based on an improved epsilon constraint-handling method. Firstly, MODE-SaE self-adaptively adjusts the epsilon level in line with the maximum and minimum constraint violation values of infeasible individuals. It can prevent epsilon level setting from being unreasonable. Then, the feasible solutions are saved to the external archive and take part in the population evolution by a co-evolution strategy. Finally, MODE-SaE switches the global search and local search by self-switching parameters of search engine to balance the convergence and distribution. With the aim of evaluating the performance of MODE-SaE, a real-world problem with low feasible ratio in decision space and fourteen bench-mark test problems, are used to test MODE-SaE and five other state-of-the-art constrained multi-objective evolution algorithms. The experimental results fully demonstrate the superiority of MODE-SaE on all mentioned test problems, which indicates the effectiveness of the proposed algorithm for CMOPs which have low feasible ratio in search space.     

面向复杂约束优化问题的进化算法综述

约束优化是多数实际工程应用优化问题的呈现方式.进化算法由于其高效的表现,近年来被广泛应用于约束优化问题求解.但约束条件使得问题解空间离散、缩小、改变,给进化算法求解约束优化问题带来极大挑战.在此背景下,融合约束处理技术的进化算法成为研究热点.此外,随着研究的深入,近年来约束处理技术在复杂工程应用问题优化中得到了广泛发展,例如多目标、高维、等式优化等.根据复杂性的缘由,将面向复杂约束优化问题的进化优化分为面向复杂目标的进化约束优化算法和面向复杂约束场景的进化算法两种类别进行综述,其中,重点探讨了实际工程应用的复杂性对约束处理技术的挑战和目前研究的最新进展,并最后总结了未来的研究趋势与挑战.     

Coevolutionary particle swarm optimization using Gaussian distribution for solving constrained optimization problems.

In this correspondence, an approach based on coevolutionary particle swarm optimization to solve constrained optimization problems formulated as min-max problems is presented. In standard or canonical particle swarm optimization (PSO), a uniform probability distribution is used to generate random numbers for the accelerating coefficients of the local and global terms. We propose a Gaussian probability distribution to generate the accelerating coefficients of PSO. Two populations of PSO using Gaussian distribution are used on the optimization algorithm that is tested on a suite of well-known benchmark constrained optimization problems. Results have been compared with the canonical PSO (constriction factor) and with a coevolutionary genetic algorithm. Simulation results show the suitability of the proposed algorithm in terms of effectiveness and robustness.     

A coevolutionary differential evolution with harmony search for reliability-redundancy optimization

Solving reliability-redundancy optimization problems via meta-heuristic algorithms has attracted increasing attention in recent years. In this paper, an effective coevolutionary differential evolution with harmony search algorithm (CDEHS) is proposed to solve the reliability-redundancy optimization problem by dividing the problem into a continuous part and an integer part. In CDEHS, two populations evolve simultaneously and cooperatively, where one population for the continuous part evolves by means of differential evolution while another population for the integer part evolves by means of harmony search. After half of the whole evolving process, the integer part stops evolving and provides the best solution to the other part for further evolving with differential evolution. Simulations results based on three typical problems and comparisons with some existing algorithms show that the proposed CDEHS is effective, efficient and robust for solving the reliability-redundancy optimization problem.     

A distributed Cooperative coevolutionary algorithm for multiobjective optimization

Recent advances in evolutionary algorithms show that coevolutionary architectures are effective ways to broaden the use of traditional evolutionary algorithms. This paper presents a cooperative coevolutionary algorithm (CCEA) for multiobjective optimization, which applies the divide-and-conquer approach to decompose decision vectors into smaller components and evolves multiple solutions in the form of cooperative subpopulations. Incorporated with various features like archiving, dynamic sharing, and extending operator, the CCEA is capable of maintaining archive diversity in the evolution and distributing the solutions uniformly along the Pareto front. Exploiting the inherent parallelism of cooperative coevolution, the CCEA can be formulated into a distributed cooperative coevolutionary algorithm (DCCEA) suitable for concurrent processing that allows inter-communication of subpopulations residing in networked computers, and hence expedites the computational speed by sharing the workload among multiple computers. Simulation results show that the CCEA is competitive in finding the tradeoff solutions, and the DCCEA can effectively reduce the simulation runtime without sacrificing the performance of CCEA as the number of peers is increased.     

基于内部罚函数的进化算法求解约束优化问题

为解决现有约束处理方法可行解的适应度函数不包含约束条件的问题,提出了一种内部罚函数候选解筛选规则.该候选解筛选规则分别对可行解和不可行解采用内部罚函数和约束违反度进行筛选,从而达到平衡最小化目标函数和满足约束条件的目的.以进化策略算法为基础,给出了基于内部罚函数候选解筛选规则的进化算法的一个实现.进一步地,从理论和实验角度分别验证了内部罚函数候选解筛选规则的有效性:以(1+1)进化算法为例,从进化成功率方面验证了内部罚函数候选解筛选规则的理论有效性;通过13个测试问题的数值实验,从进化成功率、候选解后代是可行解的比例、进化步长和收敛速度方面验证了内部罚函数候选解筛选规则的实验有效性.     

自收缩性的演化算法及其在布局优化问题中的应用

布局优化问题是现代工程应用中广泛存在的一类组合优化问题,但在理论上它却属于NPC(NP-Complete)问题,如果需考虑性能约束,则问题将更难于求解。论文基于演化算法自适应,自组织,自学习的特性,针对布局优化问题自身的特点,提出了一种自收缩性的演化算法(SCEA)。该算法采用浮点编码方式,定义了二元实向量类型的适应值及适应值间的严格偏序关系。算法借鉴日常生活中的一个简单事实—振动容器则装物更多,引入了三类自适应性的收缩算子(其中第三类特别适用于带性能约束的布局优化问题)。此外,文中使用了对带约束的函数优化问题特别有效的多父体杂交算子,并且针对带性能约束的布局优化问题,提出了“零性能约束初始化”过程。文后,引用了两个带性能约束的布局优化问题的已知例子和一个作者构造的较大规模布局优化问题的例子,实验结果表明,前两个问题对比目前已知最好结果无论在求解时间或结果的精度上均有较大突破,后一个问题也获得了相当好的结果,从而充分验证了算法的有效性和可行性。     

约束进化算法及其应用研究综述

约束优化问题广泛存在于科学研究和工程实践中,其对应的约束优化进化算法也成为了进化领域的重要研究方向。约束优化进化算法的本质问题是如何有效地利用不可行解和可行解的信息,平衡目标函数和约束条件,使得算法更加高效。首先对约束优化问题进行定义;然后详细分析了目前主流的约束进化算法,同时,基于不同的约束处理机制,将这些机制分为约束和目标分离法、惩罚函数法、多目标优化法、混合法和其他算法,并对这些方法进行了详细的分析和总结;接着指出约束进化算法亟待解决的问题,并明确指出未来需要进一步研究的方向;最后对约束进化算法在工程优化、电子和通信工程、机械设计、环境资源配置、科研领域和管理分配等方面的应用进行了介绍。     

A hybrid cooperative search algorithm for constrained optimization

Many engineering design problems can be formulated as constrained optimization problems which often consist of many mixed equality and inequality constraints. In this article, a hybrid coevolutionary method is developed to solve constrained optimization problems formulated as min–max problems. The new method is fast and capable of global search because of combining particle swarm optimization and gradient search to balance exploration and exploitation. It starts by transforming the problem into unconstrained one using an augmented Lagrangian function, then using two groups to optimize different components of the solution vector in a cooperative procedure. In each group, the final stage of the search procedure is accelerated by via a simple local search method on the best point reached by the preceding exploration based search. We validated the effectiveness and robustness of the proposed algorithm using several engineering problems taken from the specialised literature.