Introducing robustness in multi-objective optimization

In optimization studies including multi-objective optimization, the main focus is placed on finding the global optimum or global Pareto-optimal solutions, representing the best possible objective values. However, in practice, users may not always be interested in finding the so-called global best solutions, particularly when these solutions are quite sensitive to the variable perturbations which cannot be avoided in practice. In such cases, practitioners are interested in finding the robust solutions which are less sensitive to small perturbations in variables. Although robust optimization is dealt with in detail in single-objective evolutionary optimization studies, in this paper, we present two different robust multi-objective optimization procedures, where the emphasis is to find a robust frontier, instead of the global Pareto-optimal frontier in a problem. The first procedure is a straightforward extension of a technique used for single-objective optimization and the second procedure is a more practical approach enabling a user to set the extent of robustness desired in a problem. To demonstrate the differences between global and robust multi-objective optimization principles and the differences between the two robust optimization procedures suggested here, we develop a number of constrained and unconstrained test problems having two and three objectives and show simulation results using an evolutionary multi-objective optimization (EMO) algorithm. Finally, we also apply both robust optimization methodologies to an engineering design problem.     

遗传算法GA结合BFGS预测玻璃组成

复杂组成玻璃的定量设计过程往往涉及复杂的多目标优化问题，依照传统方法逐个实现目标性质设计，常常顾此失彼，玻璃开发周期很长，为此将多个性质加权平均后转化为一个综合指数，通过对综合指数的模拟来实现多个目标的优化，而这一过程常常需要强劲的优化算法，对于这种变量多、函数关系复杂的优化问题，传统的基于梯度的算法常常过早收敛于局部最优，适合全局搜索的遗传算法的局部精细搜索能力不强，因此将遗传算法与BFGS算法结合用于玻璃陶瓷复合材料组分设计，弥补了二者的缺点，实现了玻璃组成快速推确的程序设计，作为一种通用算法，GA-BFGS算法亦可用于其它最优化过程。     

基于改进初始解的遗传算法的布局设计方法

带性能约束的三维布局问题属于具有很强应用背景的组合优化问题,进行了基于全局的布局求解方法的探索。由于NP完全问题的计算复杂性,使得遗传算法求解问题的全局最优解时效率较低。改进了遗传算法的初始解,对提高算法的效率进行了研究。并以旋转卫星舱布局的简化模型为背景,建立了多目标优化数学模型。实例结果与传统遗传算法以及乘子法的计算结果比较,表明该算法具有较好的求解效率。     

Immunity-based hybrid evolutionary algorithm for multi-objective optimization in global container repositioning

The development of evolutionary algorithms for optimization has always been a stimulating and growing research area with an increasing demand in using them to solve complex industrial optimization problems. A novel immunity-based hybrid evolutionary algorithm known as Hybrid Artificial Immune Systems (HAIS) for solving both unconstrained and constrained multi-objective optimization problems is developed in this research. The algorithm adopts the clonal selection and immune suppression theories, with a sorting scheme featuring uniform crossover, multi-point mutation, non-dominance and crowding distance sorting to attain the Pareto optimal front in an efficient manner. The proposed algorithm was verified with nine benchmarking functions on its global optimal search ability as well as compared with four optimization algorithms to assess its diversity and spread. Sensitivity analysis was also carried out to investigate the selection of key parameters of the algorithm. It is found that the developed immunity-based hybrid evolutionary algorithm provides a useful means for solving optimization problems and has successfully applied to the problem of global repositioning of containers, which is one of a constrained multi-objective optimization problem. The developed HAIS will assist shipping liners on timely decision making and planning of container repositioning operations in global container transportation business in an optimized and cost effective manner.     

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.     

一种基于正交设计的快速差分演化算法及其应用研究

为了进一步加快差分演化算法的速度和增强算法的鲁棒性,提出了一种基于正交设计的快速差分演化算法,并把它应用于函数优化问题的求解中.新算法在保持传统差分演化算法的简单、有效等特性的同时,具有以下特征：1）采用基于正交设计的杂交算子,并结合直观统计法产生最优子个体;2）采用决策变量分块策略,以减少正交实验次数,加快算法收敛速度;3）提出一种基于非凸理论的多父体混合自适应杂交变异算子,以增强算法的非凸搜索能力和自适应能力;4）简化基本差分演化算法的缩放因子,尽量减少算法的控制参数,方便工程人员的使用.通过对12个标准测试函数进行实验,并与其他演化算法的结果相比较,其结果表明,新算法在解的精度、稳定性和收敛性上表现出很好的性能.     

Finite element based hybrid evolutionary optimization approach to solving rigid pavement inversion problem

This paper focuses on the development of a new backcalculation method for concrete road structures based on a hybrid evolutionary global optimization algorithm, namely shuffled complex evolution (SCE). Evolutionary optimization algorithms are ideally suited for intrinsically multi-modal, non-convex, and discontinuous real-world problems such as pavement backcalculation because of their ability to explore very large and complex search spaces and locate the globally optimal solution using a parallel search mechanism as opposed to a point-by-point search mechanism employed by traditional optimization algorithms. SCE, a type of evolutionary optimization algorithms based on the tradeoff of exploration and exploitation, has proved to be an efficient method for many global optimization problems and in some cases it does not suffer the difficulties encountered by other evolutionary computation techniques. The SCE optimization approach is hybridized with a neural networks surrogate finite-element based forward pavement response model to enable rapid computation of global or near-global pavement layer moduli solutions. The proposed rigid pavement backcalculation model is evaluated using field non-destructive test data acquired from a full-scale airport pavement test facility.     

基于自适应权重和莱维飞行的改进海鸥优化算法

在齿轮系设计问题中, 传统算法存在计算复杂与精度低等缺点, 海鸥优化算法(SOA)得益于其算法原理简单、通用性强、参数少等特性, 现多用于工程设计问题. 然而, 标准海鸥优化算法易出现寻优精度低、搜索速度慢等问题, 本文提出一种混合策略改进的海鸥优化算法(WLSOA). 首先, 利用非线性递减策略增强海鸥优化算法的探索开发能力, 提高寻优精度. 其次, 在海鸥攻击阶段引入自适应权重平衡全局与局部的搜索能力和加入莱维飞行步长对当前最优解进行扰动, 提高算法跳出局部最优值的能力. 然后分别使用WLSOA、黄金正弦算法、鲸鱼优化算法、粒子群优化算法、传统海鸥优化算法及最新提出的改进海鸥优化算法, 通过在9个经典的测试函数上进行仿真实验来探究WLSOA的性能. 结果表明, WLSOA比其他6种算法寻优精度更高, 收敛速度更快. 最后, 在齿轮系设计问题上, 通过与其他13种常见的群智能算法的比较表明, WLSOA的求解性能优于其他算法.     

Multimodal optimization using a bi-objective evolutionary algorithm

In a multimodal optimization task, the main purpose is to find multiple optimal solutions (global and local), so that the user can have better knowledge about different optimal solutions in the search space and as and when needed, the current solution may be switched to another suitable optimum solution. To this end, evolutionary optimization algorithms (EA) stand as viable methodologies mainly due to their ability to find and capture multiple solutions within a population in a single simulation run. With the preselection method suggested in 1970, there has been a steady suggestion of new algorithms. Most of these methodologies employed a niching scheme in an existing single-objective evolutionary algorithm framework so that similar solutions in a population are deemphasized in order to focus and maintain multiple distant yet near-optimal solutions. In this paper, we use a completely different strategy in which the single-objective multimodal optimization problem is converted into a suitable bi-objective optimization problem so that all optimal solutions become members of the resulting weak Pareto-optimal set. With the modified definitions of domination and different formulations of an artificially created additional objective function, we present successful results on problems with as large as 500 optima. Most past multimodal EA studies considered problems having only a few variables. In this paper, we have solved up to 16-variable test problems having as many as 48 optimal solutions and for the first time suggested multimodal constrained test problems which are scalable in terms of number of optima, constraints, and variables. The concept of using bi-objective optimization for solving single-objective multimodal optimization problems seems novel and interesting, and more importantly opens up further avenues for research and application.     

Balance between genetic search and local search in memetic algorithms for multiobjective permutation flowshop scheduling

This paper shows how the performance of evolutionary multiobjective optimization (EMO) algorithms can be improved by hybridization with local search. The main positive effect of the hybridization is the improvement in the convergence speed to the Pareto front. On the other hand, the main negative effect is the increase in the computation time per generation. Thus, the number of generations is decreased when the available computation time is limited. As a result, the global search ability of EMO algorithms is not fully utilized. These positive and negative effects are examined by computational experiments on multiobjective permutation flowshop scheduling problems. Results of our computational experiments clearly show the importance of striking a balance between genetic search and local search. In this paper, we first modify our former multiobjective genetic local search (MOGLS) algorithm by choosing only good individuals as initial solutions for local search and assigning an appropriate local search direction to each initial solution. Next, we demonstrate the importance of striking a balance between genetic search and local search through computational experiments. Then we compare the modified MOGLS with recently developed EMO algorithms: the strength Pareto evolutionary algorithm and revised nondominated sorting genetic algorithm. Finally, we demonstrate that a local search can be easily combined with those EMO algorithms for designing multiobjective memetic algorithms.     

全局与局部模型交替辅助的差分进化算法

为求解实际复杂工程应用中的高维计算费时优化问题,提出一种全局与局部代理模型交替辅助的差分进化算法。利用历史样本训练全局和局部代理模型,通过交替搜索全局和局部代理模型得到模型最优解并对其进行真实目标函数评价,实现探索和开采的平衡以减少真实目标函数的计算次数,同时通过针对性地选择个体进行真实目标函数计算,辅助算法快速找到目标函数的较优解。在15个低维测试问题和14个高维测试问题上的实验结果表明,在有限的计算资源情况下,该算法在12个低维测试问题上相较于最优重启策略代理辅助的社会学习粒子群优化算法、基于主动学习的代理模型辅助的粒子群优化算法等表现更好,在7个高维测试问题上相较于高斯过程辅助的进化算法、代理模型辅助的分层粒子群优化算法、求解高维费时问题的代理辅助的多种群优化算法等能找到目标函数的更优解。     

A parallel double-level multiobjective evolutionary algorithm for robust optimization

Robust optimization is a popular method to tackle uncertain optimization problems. However, traditional robust optimization can only find a single solution in one run which is not flexible enough for decision-makers to select a satisfying solution according to their preferences. Besides, traditional robust optimization often takes a large number of Monte Carlo simulations to get a numeric solution, which is quite time-consuming. To address these problems, this paper proposes a parallel double-level multiobjective evolutionary algorithm (PDL-MOEA). In PDL-MOEA, a single-objective uncertain optimization problem is translated into a bi-objective one by conserving the expectation and the variance as two objectives, so that the algorithm can provide decision-makers with a group of solutions with different stabilities. Further, a parallel evolutionary mechanism based on message passing interface (MPI) is proposed to parallel the algorithm. The parallel mechanism adopts a double-level design, i.e., global level and sub-problem level. The global level acts as a master, which maintains the global population information. At the sub-problem level, the optimization problem is decomposed into a set of sub-problems which can be solved in parallel, thus reducing the computation time. Experimental results show that PDL-MOEA generally outperforms several state-of-the-art serial/parallel MOEAs in terms of accuracy, efficiency, and scalability.     

Constrained self-adaptive differential evolution based design of robust optimal fixed structure controller

This paper presents a constrained Self-adaptive Differential Evolution (SaDE) algorithm for the design of robust optimal fixed structure controllers with uncertainties and disturbance. Almost all real world optimization problems have constraints which should be satisfied along with the best optimal solution for the problem. In evolutionary algorithms (EAs) the presence of constraints reduces the feasible region and complicates the search process. Therefore, a suitable method to handle the constraints must also be executed. In the SaDE algorithm, four mutation strategies and the control parameter CR are self-adapted. Self-adaptive Penalty (SP) method is introduced into the SaDE algorithm for constraint handling. The performance of SaDE algorithm is demonstrated on the design of robust optimal fixed structure controller of three systems, namely the linearized magnetic levitation system, F-8 aircraft linearized model and a SISO plant. For the comparison purpose, reported results of constrained PSO algorithm and five DE algorithms with different strategies and parameter values are taken into account. Statistical performance in 20 independent runs is considered to compare the performance of algorithms. From the obtained results, it is observed that SaDE algorithm is able to self-adapt the mutation strategy and the crossover rate and hence performs better than the other variants of DE and the constrained PSO algorithm. Better performance of SaDE is achieved by sustained maintenance of diversity throughout the evolutionary process thus producing better individuals consistently. This also aids the algorithm to escape from local optima thereby avoiding premature convergence.     

组织进化数值优化算法

基于经济学中“组织”的概念 ,该文提出一种新的进化算法———组织进化算法 ,来解决无约束和有约束的数值优化问题 .该算法与传统遗传算法、进化规划、进化策略的运行机制完全不同 ,其进化操作不直接作用于个体上 ,而作用在组织上 ,为此 ,该文定义了三种组织进化算子———分裂算子、吞并算子和合作算子来引导种群进化 .理论分析证明组织进化算法具有全局收敛性 .实验中 ,用 4个无约束和 6个有约束标准函数对算法进行了测试 ,与 3个新算法作了比较 ,并对组织进化算法的性能作了深入分析 .结果表明 ,该文算法无论在解的质量上还是在计算复杂度上都优于其它算法 .对于有约束问题 ,只用了简单的静态罚函数就得到了良好的效果 ,这表明该文算法的搜索机制非常有效 ,不易陷入局部最优 .最后 ,参数分析的结果表明该文算法具有性能稳定、成功率高、对参数不敏感等优越的性能     

Hybrid evolutionary algorithm and application to structural optimization

This paper introduces a new evolutionary algorithm with a globally stochastic but locally heuristic search strategy. It is implemented by incorporating a modified micro-genetic algorithm with two local optimization operators. Performance tests using two benchmarking functions demonstrate that the new algorithm has excellent convergence performance when applied to multimodal optimization problems. The number of objective function evaluations required to obtain global optima is only 3.5–3.7% of that of using the conventional micro-genetic algorithm. The new algorithm is used to optimize the design of an 18-bar truss, with the aim of minimizing its weight while meeting the stress, section area, and geometry constraints. The corresponding optimal design is obtained with considerably fewer computational operations than required for the existing algorithms.     

A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice

Most engineering optimization algorithms are based on numerical linear and nonlinear programming methods that require substantial gradient information and usually seek to improve the solution in the neighborhood of a starting point. These algorithms, however, reveal a limited approach to complicated real-world optimization problems. If there is more than one local optimum in the problem, the result may depend on the selection of an initial point, and the obtained optimal solution may not necessarily be the global optimum. This paper describes a new harmony search (HS) meta-heuristic algorithm-based approach for engineering optimization problems with continuous design variables. This recently developed HS algorithm is conceptualized using the musical process of searching for a perfect state of harmony. It uses a stochastic random search instead of a gradient search so that derivative information is unnecessary. Various engineering optimization problems, including mathematical function minimization and structural engineering optimization problems, are presented to demonstrate the effectiveness and robustness of the HS algorithm. The results indicate that the proposed approach is a powerful search and optimization technique that may yield better solutions to engineering problems than those obtained using current algorithms.     

基于浮点型编码策略的差分多目标柔性车间调度优化

运用进化算法求解柔性车间调度问题时,编码的特殊性对进化策略造成的局限制约了算法的搜索能力。为此,提出一种基于浮点型编码策略的差分多目标优化算法。该算法采用基于工序权重的浮点数编码—解码机制,消除了排列组合型编码方式对进化操作带来的约束,运用差分进化策略生成新个体,以提高优秀个体产生的几率,进而保证算法有更好的收敛性。将算法与传统算法及其改进形式在相同测试用例上进行对比,结果表明,本算法在保证收敛性的同时,搜索到更多的非支配个体,体现出更好的分布性。此外,提出了平行决策和等价平行决策的定义,将柔性车间调度模型的研究拓展至决策空间。     

基于DE 和SA 的Memetic 高维全局优化算法

针对高维复杂多模态优化问题,传统的进化算法存在收敛速度慢,求解精度低等缺点,提出一种面向高维优化问题的Memetic全局优化算法。算法通过全局搜索和局部搜索结合的混合搜索策略,采用多模式并行差分进化算法进行全局搜索,基于高斯分布估计的模拟退火算法进行局部搜索。改进后的Memetic算法不仅继承了差分进化算法能发现全局最优解的优点,而且能大幅度提高搜索效率。最后,通过对4个高维多峰值Benchmark函数进行仿真实验,实验结果表明本文算法有效提高了算法的收敛速度和求解精度。     

Truss optimization on shape and sizing with frequency constraints based on parallel genetic algorithm

Truss shape and sizing optimization under frequency constraints is extremely useful when improving the dynamic performance of structures. However, coupling of two different types of design variables, nodal coordinates and cross-sectional areas, often lead to slow convergence or even divergence. Because shape and sizing variables coupled increase the number of design variables and the changes of shape and sizing variables are of widely different orders of magnitude. Otherwise, multiple frequency constraints often cause difficult dynamic sensitivity analysis. Thus optimal criteria and mathematical programming methods have considerable limitations on solving the problems because of needing complex dynamic sensitivity analysis and being easily trapped into the local optima. Genetic Algorithms (GAs) show great potentials to solve the truss shape and sizing optimization problems. Since GAs adopt global probabilistic population search techniques and require no gradient information. The improved genetic algorithms can effectively increase the solution quality. However, the serial GA is computationally expensive and is limited on gaining higher quality solutions. To solve the truss shape and sizing optimization problems with frequency constraints more effectively and efficiently, a Niche Hybrid Parallel Genetic Algorithm (NHPGA) is proposed to significantly reduce the computational cost and to further improve solution quality. The NHPGA is to blend the advantages of parallel computing, simplex search and genetic algorithm with niche technique. Several typical truss optimization examples demonstrate that NHPGA can significantly reduce computing time and attain higher quality solutions. It also suggests that the NHPGA provide a potential algorithm architecture, which effectively combines the robust and global search characteristics of genetic algorithm, strong exploitation ability of simplex search and computational speedup property of parallel computing.     

Comparison of Stochastic Global Optimization Methods to Estimate Neural Network Weights

Training a neural network is a difficult optimization problem because of numerous local minima. Many global search algorithms have been used to train neural networks. However, local search algorithms are more efficient with computational resources, and therefore numerous random restarts with a local algorithm may be more effective than a global algorithm. This study uses Monte-Carlo simulations to determine the efficiency of a local search algorithm relative to nine stochastic global algorithms when using a neural network on function approximation problems. The computational requirements of the global algorithms are several times higher than the local algorithm and there is little gain in using the global algorithms to train neural networks. Since the global algorithms only marginally outperform the local algorithm in obtaining a lower local minimum and they require more computational resources, the results in this study indicate that with respect to the specific algorithms and function approximation problems studied, there is little evidence to show that a global algorithm should be used over a more traditional local optimization routine for training neural networks. Further, neural networks should not be estimated from a single set of starting values whether a global or local optimization method is used.