Vehicle power train optimization using multi-objective bird swarm algorithm

In this paper, a multi-objective bird swarm algorithm (MOBSA) is proposed to cope with multi-objective optimization problems. The algorithm is explored based on BSA which is an evolutionary algorithm suitable for single objective optimization. In this paper, non-dominated sorting approach is used to distinguish optimal solutions and parallel coordinates is applied to evaluate the distribution density of non-dominated solution and further update the external archive when it is full to overflowing, which ensure faster convergence and more widespread of Pareto front. Then, the MOBSA is adopted to optimize benchmark problems. The results demonstrate that MOBSA gets better performance compared with NSGA-II and MOPSO. Since a vehicle power train problem could be treated as a typical multi-objective optimization problem with constraints, with integration of constrained non-dominated solution, MOBSA is adopted to acquire optimal gear ratios and optimize vehicle power train. The results compared with other popular algorithm prove the proposed algorithm is more suitable for constrained multi-objective optimization problem in engineering field.

     

A surrogate-assisted evolutionary algorithm based on the genetic diversity objective

In this work, a novel surrogate-assisted memetic algorithm is proposed which is based on the preservation of genetic diversity within the population. The aim of the algorithm is to solve multi-objective optimization problems featuring computationally expensive fitness functions in an efficient manner. The main novelty is the use of an evolutionary algorithm as global searcher that treats the genetic diversity as an objective during the evolution and uses it, together with a non-dominated sorting approach, to assign the ranks. This algorithm, coupled with a gradient-based algorithm as local searcher and a back-propagation neural network as global surrogate model, demonstrates to provide a reliable and effective balance between exploration and exploitation. A detailed performance analysis has been conducted on five commonly used multi-objective problems, each one involving distinct features that can make the convergence difficult toward the Pareto-optimal front. In most cases, the proposed algorithm outperformed the other state-of-the-art evolutionary algorithms considered in the comparison, assuring higher repeatability on the final non-dominated set, deeper convergence level and higher convergence rate. It also demonstrates a clear ability to widely cover the Pareto-optimal front with larger percentage of non-dominated solutions if compared to the total number of function evaluations.     

Stochastic radial basis function algorithms for large-scale optimization involving expensive black-box objective and constraint functions

This paper presents a new algorithm for derivative-free optimization of expensive black-box objective functions subject to expensive black-box inequality constraints. The proposed algorithm, called ConstrLMSRBF, uses radial basis function (RBF) surrogate models and is an extension of the Local Metric Stochastic RBF (LMSRBF) algorithm by Regis and Shoemaker (2007a) [1] that can handle black-box inequality constraints. Previous algorithms for the optimization of expensive functions using surrogate models have mostly dealt with bound constrained problems where only the objective function is expensive, and so, the surrogate models are used to approximate the objective function only. In contrast, ConstrLMSRBF builds RBF surrogate models for the objective function and also for all the constraint functions in each iteration, and uses these RBF models to guide the selection of the next point where the objective and constraint functions will be evaluated. Computational results indicate that ConstrLMSRBF is better than alternative methods on 9 out of 14 test problems and on the MOPTA08 problem from the automotive industry (Jones, 2008 [2]). The MOPTA08 problem has 124 decision variables and 68 inequality constraints and is considered a large-scale problem in the area of expensive black-box optimization. The alternative methods include a Mesh Adaptive Direct Search (MADS) algorithm (Abramson and Audet, 2006 [3]; Audet and Dennis, 2006 [4]) that uses a kriging-based surrogate model, the Multistart LMSRBF algorithm by Regis and Shoemaker (2007a) [1] modified to handle black-box constraints via a penalty approach, a genetic algorithm, a pattern search algorithm, a sequential quadratic programming algorithm, and COBYLA (Powell, 1994 [5]), which is a derivative-free trust-region algorithm. Based on the results of this study, the results in Jones (2008) [2] and other approaches presented at the ISMP 2009 conference, ConstrLMSRBF appears to be among the best, if not the best, known algorithm for the MOPTA08 problem in the sense of providing the most improvement from an initial feasible solution within a very limited number of objective and constraint function evaluations.     

Improving surrogate-assisted variable fidelity multi-objective optimization using a clustering algorithm

Surrogate-assisted evolutionary optimization has proved to be effective in reducing optimization time, as surrogates, or meta-models can approximate expensive fitness functions in the optimization run. While this is a successful strategy to improve optimization efficiency, challenges arise when constructing surrogate models in higher dimensional function space, where the trade space between multiple conflicting objectives is increasingly complex. This complexity makes it difficult to ensure the accuracy of the surrogates. In this article, a new surrogate management strategy is presented to address this problem. A k-means clustering algorithm is employed to partition model data into local surrogate models. The variable fidelity optimization scheme proposed in the author's previous work is revised to incorporate this clustering algorithm for surrogate model construction. The applicability of the proposed algorithm is illustrated on six standard test problems. The presented algorithm is also examined in a three-objective stiffened panel optimization design problem to show its superiority in surrogate-assisted multi-objective optimization in higher dimensional objective function space. Performance metrics show that the proposed surrogate handling strategy clearly outperforms the single surrogate strategy as the surrogate size increases.     

Approximate multi-objective optimization using conservative and feasible moving least squares method: application to automotive knuckle design

The original version of the moving least squares method (MLSM) does not always ensure solution feasibility for nonlinear and/or non-convex functions in the context of meta-model-based approximate optimization. The paper explores a new implementation of MLSM that ensures the conservative feasibility of Pareto optimal solutions in non-dominated sorting genetic algorithm (NSGA-II)-based approximate multi-objective optimization. We devised a ‘conservative and feasible MLSM’ (CF-MLSM) to realize the conservativeness and feasibility of multi-objective Pareto optimal solutions for both unconstrained and constrained problems. We verified the usefulness of our proposed approach by exploring strength-based sizing optimization of an automotive knuckle component under bump and brake loading constraints.     

基于NSGA-II 的具有多目标子学科的协同优化方法

针对子学科具有物理目标的多目标协同优化问题, 研究基于NSGA-II 的求解策略. 鉴于子学科个体满足约束可行性的进化过程与系统级分配期望值无关, 提出具有良好的可行性和多样性的初始种群生成方法, 以提高多目标子学科的计算效率和计算精度. 为了解决由一致性目标函数与物理目标函数的作用不同而造成的NSGA-II 非支配级排序困难, 提出将子学科一致性目标函数转化为子学科自身约束的策略. 最后, 利用工程算例对所提出方法的有效性进行了验证.

     

Pareto frontier for simultaneous structural and manufacturing optimization of a composite part

Optimized design of composite structures requires simultaneous optimization of structural performance and manufacturing process. Such a challenge calls for a multi-objective optimization. Here, a generating multi-objective optimization method called normalized normal constraint method, which attains a set of optimal solutions and allows the designer to explore design alternatives before making the final decision, is coupled with a local-global search called constrained globalized bounded Nelder–Mead method. The proposed approach is applied to the design of a Z-shaped composite bracket for optimization of structural and manufacturing objectives. Comparison of the results with non-dominated sorting genetic algorithm (NSGA-II) shows that when only a small number of function evaluations are possible and a few Pareto optima are desired, the proposed method outperforms NSGA-II in terms of convergence to the true Pareto frontier. The results are validated by an enumeration search and by exploring the neighbourhood of the final solutions.     

An efficient non-dominated sorting method for evolutionary algorithms

We present a new non-dominated sorting algorithm to generate the non-dominated fronts in multi-objective optimization with evolutionary algorithms, particularly the NSGA-II. The non-dominated sorting algorithm used by NSGA-II has a time complexity of O(MN(2)) in generating non-dominated fronts in one generation (iteration) for a population size N and M objective functions. Since generating non-dominated fronts takes the majority of total computational time (excluding the cost of fitness evaluations) of NSGA-II, making this algorithm faster will significantly improve the overall efficiency of NSGA-II and other genetic algorithms using non-dominated sorting. The new non-dominated sorting algorithm proposed in this study reduces the number of redundant comparisons existing in the algorithm of NSGA-II by recording the dominance information among solutions from their first comparisons. By utilizing a new data structure called the dominance tree and the divide-and-conquer mechanism, the new algorithm is faster than NSGA-II for different numbers of objective functions. Although the number of solution comparisons by the proposed algorithm is close to that of NSGA-II when the number of objectives becomes large, the total computational time shows that the proposed algorithm still has better efficiency because of the adoption of the dominance tree structure and the divide-and-conquer mechanism.     

SO-MI: A surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems

This paper introduces a surrogate model based algorithm for computationally expensive mixed-integer black-box global optimization problems with both binary and non-binary integer variables that may have computationally expensive constraints. The goal is to find accurate solutions with relatively few function evaluations. A radial basis function surrogate model (response surface) is used to select candidates for integer and continuous decision variable points at which the computationally expensive objective and constraint functions are to be evaluated. In every iteration multiple new points are selected based on different methods, and the function evaluations are done in parallel. The algorithm converges to the global optimum almost surely. The performance of this new algorithm, SO-MI, is compared to a branch and bound algorithm for nonlinear problems, a genetic algorithm, and the NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search) algorithm for mixed-integer problems on 16 test problems from the literature (constrained, unconstrained, unimodal and multimodal problems), as well as on two application problems arising from structural optimization, and three application problems from optimal reliability design. The numerical experiments show that SO-MI reaches significantly better results than the other algorithms when the number of function evaluations is very restricted (200–300 evaluations).     

Evolutionary trajectory planning for an industrial robot

This paper presents a novel general method for computing optimal motions of an industrial robot manipulator (AdeptOne XL robot) in the presence of fixed and oscillating obstacles. The optimization model considers the nonlinear manipulator dynamics, actuator constraints, joint limits, and obstacle avoidance. The problem has 6 objective functions, 88 variables, and 21 constraints. Two evolutionary algorithms, namely, elitist non-dominated sorting genetic algorithm (NSGA-II) and multi-objective differential evolution (MODE), have been used for the optimization. Two methods (normalized weighting objective functions and average fitness factor) are used to select the best solution tradeoffs. Two multi-objective performance measures, namely solution spread measure and ratio of non-dominated individuals, are used to evaluate the Pareto optimal fronts. Two multi-objective performance measures, namely, optimizer overhead and algorithm effort, are used to find the computational effort of the optimization algorithm. The trajectories are defined by B-spline functions. The results obtained from NSGA-II and MODE are compared and analyzed.     

A modified objective function method with feasible-guiding strategy to solve constrained multi-objective optimization problems

For constrained multi-objective optimization problems (CMOPs), how to preserve infeasible individuals and make use of them is a problem to be solved. In this case, a modified objective function method with feasible-guiding strategy on the basis of NSGA-II is proposed to handle CMOPs in this paper. The main idea of proposed algorithm is to modify the objective function values of an individual with its constraint violation values and true objective function values, of which a feasibility ratio fed back from current population is used to keep the balance, and then the feasible-guiding strategy is adopted to make use of preserved infeasible individuals. In this way, non-dominated solutions, obtained from proposed algorithm, show superiority on convergence and diversity of distribution, which can be confirmed by the comparison experiment results with other two CMOEAs on commonly used constrained test problems.     

改进的约束优化多目标遗传算法及工程应用

利用多目标法处理约束条件,提出一种改进的基于多目标优化的遗传算法用于求解约束优化问题。该算法将约束优化问题转化为两个目标的多目标优化问题; 利用庄家法构造非劣个体,将种群分为支配子种群和非支配子种群,以一定概率分别从支配子种群和非支配子种群中选择个体进行算术交叉操作,引导个体逐步向极值点靠近,增强算法的局部搜索能力,对非支配子种群进行多样性变异操作。8个标准测试函数和3个工程应用的仿真实验结果表明了该算法的有效性。     

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 RBF-based constrained global optimization algorithm for problems with computationally expensive objective and constraints

This paper presents a metamodel-based constrained optimization method, called Radial basis function-based Constrained Global Optimization (RCGO), to solve optimization problems involving computationally expensive objective function and inequality constraints. RCGO is an extension of the adaptive metamodel-based global optimization (AMGO) algorithm which can handle unconstrained black-box optimization problems. Firstly, a sequential sampling method is implemented to obtain the initial points for building the radial basis function (RBF) approximations to all computational expensive functions while enforcing a feasible solution. Then, an auxiliary objective function subject to the approximate constraints is constructed to determine the next iterative point and further improve the solution. During the process, a distance function with a group of exponents is introduced in the auxiliary function to balance the local exploitation and the global exploration. The RCGO method is tested on a series of benchmark problems, and the results demonstrate that RCGO needs fewer costly evaluations and can be applied for costly constrained problems with all infeasible start points. And the test results on the 30D problems demonstrate that RCGO has advantages in solving the problems. The proposed method is then applied to the design of a cycloid gear pump and desirable results are obtained.     

基于不可行度和内分泌原理的多目标粒子群方法

针对有约束条件的多目标优化问题,提出了一种求解带约束的基于内分泌思想的多目标粒子群算法。利用不可行度方法和约束主导原理指导进化过程中精英种群的选择操作和约束条件的处理,根据生物体激素调节机制中促激素和释放激素间的相互作用原理,考虑当前非劣解集中的个体对其最邻近的一类群体的监督控制,引入当前粒子的类全局最优位置来反映其所属类中最好位置粒子对当前粒子的影响。为验证多目标约束优化算法的有效性,对两个典型的多目标优化问题进行了仿真实验,仿真结果表明该算法能较大概率地获得多目标约束优化问题的可行Pareto最优解。     

Handling expensive multi-objective optimization problems with a cluster-based neighborhood regression model

This paper gives attention to multi-objective optimization in scenarios where objective function evaluation is expensive, that is, expensive multi-objective optimization. We firstly propose a cluster-based neighborhood regression model, which incorporates the linear regression technique to predict the descent direction and generate new potential offspring. Combining this model with the classical decomposition-based multi-objective optimization framework, we propose an efficient and effective algorithm for tackling computationally expensive multi-objective optimization problems. As opposed to the conventional approach of replacing the original time-consuming objective functions with the approximated ones obtained by surrogate model, the proposed algorithm incorporates the proposed regression model to serve as an operator producing higher-quality offspring so that the algorithm requires fewer iterations to reach a given solution quality. The proposed algorithm is compared with several state-of-the-art surrogate-assisted algorithms on a variety of well-known benchmark problems. Empirical results demonstrate that the proposed algorithm outperforms or is competitive with other peer algorithms, and has the ability to keep a good trade-off between solution quality and running time within a fairly small number of function evaluations. In particular, our proposed algorithm shows obvious superiority in terms of the computational time used for the algorithm components, and can obtain acceptable solutions for expensive problems with high efficiency.     

A fast and elitist multiobjective genetic algorithm: NSGA-II

Multi-objective evolutionary algorithms (MOEAs) that use non-dominated sorting and sharing have been criticized mainly for: (1) their O(MN³) computational complexity (where M is the number of objectives and N is the population size); (2) their non-elitism approach; and (3) the need to specify a sharing parameter. In this paper, we suggest a non-dominated sorting-based MOEA, called NSGA-II (Non-dominated Sorting Genetic Algorithm II), which alleviates all of the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN²) computational complexity is presented. Also, a selection operator is presented that creates a mating pool by combining the parent and offspring populations and selecting the best N solutions (with respect to fitness and spread). Simulation results on difficult test problems show that NSGA-II is able, for most problems, to find a much better spread of solutions and better convergence near the true Pareto-optimal front compared to the Pareto-archived evolution strategy and the strength-Pareto evolutionary algorithm - two other elitist MOEAs that pay special attention to creating a diverse Pareto-optimal front. Moreover, we modify the definition of dominance in order to solve constrained multi-objective problems efficiently. Simulation results of the constrained NSGA-II on a number of test problems, including a five-objective, seven-constraint nonlinear problem, are compared with another constrained multi-objective optimizer, and the much better performance of NSGA-II is observed     

A comparative study of local search within a surrogate-assisted multi-objective memetic algorithm framework for expensive problems

A comparative study of the impacts of various local search methodologies for the surrogate-assisted multi-objective memetic algorithm (MOMA) is presented in this paper. The base algorithm for the comparative study is the single surrogate-assisted MOMA (SS-MOMA) with the main aim being to solve expensive problems with a limited computational budget. In addition to the standard weighted sum (WS) method used in the original SS-MOMA, we studied the capabilities of other local search methods based on the achievement scalarizing function (ASF), Chebyshev function, and random mutation hill climber (RMHC) in various test problems. Several practical aspects, such as normalization and constraint handling, were also studied and implemented to deal with real-world problems. Results from the test problems showed that, in general, the SS-MOMA with ASF and Chebyshev functions was able to find higher-quality solutions that were more robust than those found with WS or RMHC; although on problems with more complicated Pareto sets SS-MOMA-WS appeared as the best. SS-MOMA-ASF in conjunction with the Chebyshev function was then tested on an airfoil-optimization problem and compared with SS-MOMA-WS and the non-dominated sorting based genetic algorithm-II (NSGA-II). The results from the airfoil problem clearly showed that SS-MOMA with an achievement-type function could find more diverse solutions than SS-MOMA-WS and NSGA-II. This suggested that for real-world applications, higher-quality solutions are more likely to be found when the surrogate-based memetic optimizer is equipped with ASF or a Chebyshev function than with other local search methods.     

异构集成代理辅助多目标粒子群优化算法

针对代理辅助进化算法在减少昂贵适应度评估时难以通过少量样本点构造高质量代理模型的问题,提出异构集成代理辅助多目标粒子群优化算法。该方法通过使用加权平均法将Kriging模型和径向基函数网络模型组合成高精度的异构集成模型,达到增强算法处理不确定性信息能力的目的。基于集成学习的两种代理模型分别应用于全局搜索和局部搜索,在多目标粒子群优化算法框架基础上,新提出的方法为每个目标函数自适应地构造了异构集成模型,利用其模型的非支配解来指导粒子群的更新,得出目标函数的最优解集。实验结果表明,所提方法提高了代理模型的搜索能力,减少了评估次数,并且随着搜索维度的增加,其计算复杂性也具有更好的可扩展性。