A new decomposition based evolutionary algorithm with uniform designs for many-objective optimization

For many-objective optimization problems, how to get a set of solutions with good convergence and diversity is a difficult and challenging work. In this paper, a new decomposition based evolutionary algorithm with uniform designs is proposed to achieve the goal. The proposed algorithm adopts the uniform design method to set the weight vectors which are uniformly distributed over the design space, and the size of the weight vectors neither increases nonlinearly with the number of objectives nor considers a formulaic setting. A crossover operator based on the uniform design method is constructed to enhance the search capacity of the proposed algorithm. Moreover, in order to improve the convergence performance of the algorithm, a sub-population strategy is used to optimize each sub-problem. Comparing with some efficient state-of-the-art algorithms, e.g., NSGAII-CE, MOEA/D and HypE, on six benchmark functions, the proposed algorithm is able to find a set of solutions with better diversity and convergence.     

On the effect of reference point in MOEA/D for multi-objective optimization

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) has continuously proven effective for multi-objective optimization. So far, the effect of weight vectors and scalarizing methods in MOEA/D has been intensively studied. However, the reference point which serves as the starting point of reference lines (determined by weight vectors) is yet to be well studied. This study aims to fill in this research gap. Ideally, the ideal point of a multi-objective problem could serve as the reference point, however, since the ideal point is often unknown beforehand, the reference point has to be estimated (or specified). In this study, the effect of the reference point specified in three representative manners, i.e., pessimistic, optimistic and dynamic (from optimistic to pessimistic), is examined on three sets of benchmark problems. Each set of the problems has different degrees of difficulty in convergence and spread. Experimental results show that (i) the reference point implicitly impacts the convergence and spread performance of MOEA/D; (ii) the pessimistic specification emphasizes more of exploiting existing regions and the optimistic specification emphasizes more of exploring new regions; (iii) the dynamic specification can strike a good balance between exploitation and exploration, exhibiting good performance for most of the test problems, and thus, is commended to use for new problems.     

A population-based algorithm for solving linear assignment problems with two objectives

The paper presents a population-based algorithm for computing approximations of the efficient solution set for the linear assignment problem with two objectives. This is a multiobjective metaheuristic based on the intensive use of three operators – a local search, a crossover and a path-relinking – performed on a population composed only of elite solutions. The initial population is a set of feasible solutions, where each solution is one optimal assignment for an appropriate weighted sum of two objectives. Genetic information is derived from the elite solutions, providing a useful genetic heritage to be exploited by crossover operators. An upper bound set, defined in the objective space, provides one acceptable limit for performing a local search. Results reported using referenced data sets have shown that the heuristic is able to quickly find a very good approximation of the efficient frontier, even in situation of heterogeneity of objective functions. In addition, this heuristic has two main advantages. It is based on simple easy-to-implement principles, and it does not need a parameter tuning phase.     

2D decision-making for multicriteria design optimization

The high dimensionality encountered in engineering design optimization due to large numbers of performance criteria and specifications leads to cumbersome and sometimes unachievable trade-off analyses. To facilitate those analyses and enhance decision-making and design selection, we propose to decompose the original problem by considering only pairs of criteria at a time, thereby making trade-off evaluation the simplest possible. For the final design integration, we develop a novel coordination mechanism that guarantees that the selected design is also preferred for the original problem. The solution of an overall large-scale problem is therefore reduced to solving a family of bicriteria subproblems and allows designers to effectively use decision-making in merely two dimensions for multicriteria design optimization.

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Margaret M. Wiecek is on leave from the Department of Mathematical Sciences, Clemson University, South Carolina 29634, USA.     

Reliability-based robust assessment for multiobjective optimization design of improving occupant restraint system performance

Optimal performance of vehicle occupant restraint system (ORS) requires an accurate assessment of occupant injury values including head, neck and chest responses, etc. To provide a feasible framework for incorporating occupant injury characteristics into the ORS design schemes, this paper presents a reliability-based robust approach for the development of the ORS. The uncertainties of design variables are addressed and the general formulations of reliable and robust design are given in the optimization process. The ORS optimization is a highly nonlinear and large scale problem. In order to save the computational cost, an optimal sampling strategy is applied to generate sample points at the stage of design of experiment (DOE). Further, to efficiently obtain a robust approximation, the support vector regression (SVR) is suggested to construct the surrogate model in the vehicle ORS design process. The multiobjective particle swarm optimization (MPSO) algorithm is used for obtaining the Pareto optimal set with emphasis on resolving conflicting requirements from some of the objectives and the Monte Carlo simulation (MCS) method is applied to perform the reliability and robustness analysis. The differences of three different Pareto fronts of the deterministic, reliable and robust multiobjective optimization designs are compared and analyzed in this study. Finally, the reliability-based robust optimization result is verified by using sled system test. The result shows that the proposed reliability-based robust optimization design is efficient in solving ORS design optimization problems.     

A new chaotic multi-verse optimization algorithm for solving engineering optimization problems

Multi-verse optimization algorithm (MVO) is one of the recent meta-heuristic optimization algorithms. The main inspiration of this algorithm came from multi-verse theory in physics. However, MVO like most optimization algorithms suffers from low convergence rate and entrapment in local optima. In this paper, a new chaotic multi-verse optimization algorithm (CMVO) is proposed to overcome these problems. The proposed CMVO is applied on 13 benchmark functions and 7 well-known design problems in the engineering and mechanical field; namely, three-bar trust, speed reduce design, pressure vessel problem, spring design, welded beam, rolling element-bearing and multiple disc clutch brake. In the current study, a modified feasible-based mechanism is employed to handle constraints. In this mechanism, four rules were used to handle the specific constraint problem through maintaining a balance between feasible and infeasible solutions. Moreover, 10 well-known chaotic maps are used to improve the performance of MVO. The experimental results showed that CMVO outperforms other meta-heuristic optimization algorithms on most of the optimization problems. Also, the results reveal that sine chaotic map is the most appropriate map to significantly boost MVO’s performance.     

基于均匀设计的支持向量机参数优化方法

在实际应用中,支持向量机的性能依赖于参数的选择。针对支持向量机的参数选择问题进行了研究和分析,提出了基于均匀设计的支持向量机参数优化方法。与基于网格搜索、粒子群算法、遗传算法等支持向量机参数优化方法进行了比较与分析,采用多个不同规模的标准的分类数据集进行测试,比较了四种方法的分类正确率和运行时间。仿真实验表明,四种方法都能找到最优参数,使支持向量机的分类正确率接近或超过分类数据集的理论精度,本文方法具有寻参时间短的特点。     

The improvement of glowworm swarm optimization for continuous optimization problems

Glowworm swarm optimization (GSO) algorithm is the one of the newest nature inspired heuristics for optimization problems. In order to enhances accuracy and convergence rate of the GSO, two strategies about the movement phase of GSO are proposed. One is the greedy acceptance criteria for the glowworms update their position one-dimension by one-dimension. The other is the new movement formulas which are inspired by artificial bee colony algorithm (ABC) and particle swarm optimization (PSO). To compare and analyze the performance of our proposed improvement GSO, a number of experiments are carried out on a set of well-known benchmark global optimization problems. The effects of the parameters about the improvement algorithms are discussed by uniform design experiment. Numerical results reveal that the proposed algorithms can find better solutions when compared to classical GSO and other heuristic algorithms and are powerful search algorithms for various global optimization problems.     

Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems

A new efficient optimization method, called ‘Teaching–Learning-Based Optimization (TLBO)’, is proposed in this paper for the optimization of mechanical design problems. This method works on the effect of influence of a teacher on learners. Like other nature-inspired algorithms, TLBO is also a population-based method and uses a population of solutions to proceed to the global solution. The population is considered as a group of learners or a class of learners. The process of TLBO is divided into two parts: the first part consists of the ‘Teacher Phase’ and the second part consists of the ‘Learner Phase’. ‘Teacher Phase’ means learning from the teacher and ‘Learner Phase’ means learning by the interaction between learners. The basic philosophy of the TLBO method is explained in detail. To check the effectiveness of the method it is tested on five different constrained benchmark test functions with different characteristics, four different benchmark mechanical design problems and six mechanical design optimization problems which have real world applications. The effectiveness of the TLBO method is compared with the other population-based optimization algorithms based on the best solution, average solution, convergence rate and computational effort. Results show that TLBO is more effective and efficient than the other optimization methods for the mechanical design optimization problems considered. This novel optimization method can be easily extended to other engineering design optimization problems.     

Shape design optimization of stationary fluid-structure interaction problems with large displacements and turbulence

This paper presents general and efficient methods for analysis and gradient based shape optimization of systems characterized as strongly coupled stationary fluid-structure interaction (FSI) problems. The incompressible fluid flow can be laminar or turbulent and is described using the Reynolds-averaged Navier-Stokes equations (RANS) together with the algebraic Baldwin–Lomax turbulence model. The structure may exhibit large displacements due to the interaction with the fluid domain, resulting in geometrically nonlinear structural behaviour and nonlinear interface coupling conditions. The problem is discretized using Galerkin and Streamline-Upwind/Petrov–Galerkin finite element methods, and the resulting nonlinear equations are solved using Newtons method. Due to the large displacements of the structure, an efficient update algorithm for the fluid mesh must be applied, leading to the use of an approximate Jacobian matrix in the solution routine. Expressions for Design Sensitivity Analysis (DSA) are derived using the direct differentiation approach, and the use of an inexact Jacobian matrix in the analysis leads to an iterative but very efficient scheme for DSA. The potential of gradient based shape optimization of fluid flow and FSI problems is illustrated by several examples.     

An intelligent moving object optimization algorithm for design problems with mixed variables, mixed constraints and multiple objectives

This paper presents an optimization algorithm for engineering design problems having a mix of continuous, discrete and integer variables; a mix of linear, non-linear, differentiable, non-differential, equality, inequality and even discontinuous design constraints; and conflicting multiple design objectives. The intelligent movement of objects (vertices and compounds) is simulated in the algorithm based on a Nelder–Mead simplex with added features to handle variable types, bound and design constraints, local optima, search initiation from an infeasible region and numerical instability, which are the common requirements for large-scale, complex optimization problems in various engineering and business disciplines. The algorithm is called an INTElligent Moving Object algorithm and tested for a wide range of benchmark problems. Validation results for several examples, which are manageable within the scope of this paper, are presented herein. Satisfactory results have been obtained for all the test problems, hence, highlighting the benefits of the proposed method.     

A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization

New challenges in engineering design lead to multiobjective (multicriteria) problems. In this context, the Pareto front supplies a set of solutions where the designer (decision-maker) has to look for the best choice according to his preferences. Visualization techniques often play a key role in helping decision-makers, but they have important restrictions for more than two-dimensional Pareto fronts. In this work, a new graphical representation, called Level Diagrams, for n-dimensional Pareto front analysis is proposed. Level Diagrams consists of representing each objective and design parameter on separate diagrams. This new technique is based on two key points: classification of Pareto front points according to their proximity to ideal points measured with a specific norm of normalized objectives (several norms can be used); and synchronization of objective and parameter diagrams. Some of the new possibilities for analyzing Pareto fronts are shown. Additionally, in order to introduce designer preferences, Level Diagrams can be coloured, so establishing a visual representation of preferences that can help the decision-maker. Finally, an example of a robust control design is presented - a benchmark proposed at the American Control Conference. This design is set as a six-dimensional multiobjective problem.     

A discrete version of particle swarm optimization for flowshop scheduling problems

Particle swarm optimization (PSO) is a novel metaheuristic inspired by the flocking behavior of birds. The applications of PSO to scheduling problems are extremely few. In this paper, we present a PSO algorithm, extended from discrete PSO, for flowshop scheduling. In the proposed algorithm, the particle and the velocity are redefined, and an efficient approach is developed to move a particle to the new sequence. To verify the proposed PSO algorithm, comparisons with a continuous PSO algorithm and two genetic algorithms are made. Computational results show that the proposed PSO algorithm is very competitive. Furthermore, we incorporate a local search scheme into the proposed algorithm, called PSO-LS. Computational results show that the local search can be really guided by PSO in our approach. Also, PSO-LS performs well in flowshop scheduling with total flow time criterion, but it requires more computation times.     

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.     

果蝇耦合均匀设计算法及其优化SVM参数

支持向量机的参数选择仍无系统的理论指导,且参数优化一直是支持向量机的一个重要研究方向。传统果蝇优化算法能够较快寻得一个较优的近似最优解,随后在该解的邻域继续迭代而造成寻优时间的严重增加。针对该问题构建了果蝇优化算法与均匀设计相耦合的果蝇耦合均匀设计算法,并将其用于支持向量机的参数优化。该算法首先利用果蝇优化算法并行寻优以快速得到所研究问题的一个较优近似最优解,然后跳转执行均匀设计的局部寻优,以获得一个更优的近似最优解。数值实验结果表明:该算法具有较快的寻优效率和较高的分类精度,验证了其在支持向量机参数优化中的有效性和可行性。     

Improving the design of sequences for DNA computing: A multiobjective evolutionary approach

Designing oligonucleotide strands that selectively hybridize to reduce undesired reactions is a critical step for successful DNA computing. To accomplish this, DNA molecules must be restricted to a wide window of thermodynamical and logical conditions, which in turn facilitate and control the algorithmic processes implemented by chemical reactions. In this paper, we propose a multiobjective evolutionary algorithm for DNA sequence design that, unlike preceding evolutionary approaches, uses a matrix-based chromosome as encoding strategy. Computational results show that a matrix-based GA along with its specific genetic operators may improve the performance for DNA sequence optimization compared to previous methods.     

Space exploration and global optimization for computationally intensive design problems: a rough set based approach

Modern engineering design problems often involve computation-intensive analysis and simulation processes. Design optimization based on such processes is desired to be efficient, informative and transparent. This work proposes a rough set based approach that can identify multiple sub-regions in a design space, within which all of the design points are expected to have a performance value equal to or less than a given level. The rough set method is applied iteratively on a growing sample set. A novel termination criterion is also developed to ensure a modest number of total expensive function evaluations to identify these sub-regions and search for the global optimum. The significance of the proposed method is twofold. First, it provides an intuitive method to establish the mapping from the performance space to the design space, i.e. given a performance level, its corresponding design region(s) can be identified. Such a mapping could be potentially used to explore and visualize the entire design space. Second, it can be naturally extended to a global optimization method. It also bears potential for more broad application to problems such as metamodeling-based design and robust design optimization. The proposed method was tested with a number of test problems and compared with a few well-known global optimization algorithms.     

A classification of methods for distributed system optimization based on formulation structure

Augmented Lagrangian coordination (ALC) is a provably convergent coordination method for multidisciplinary design optimization (MDO) that is able to treat both linking variables and linking functions (i.e. system-wide objectives and constraints). Contrary to quasi-separable problems with only linking variables, the presence of linking functions may hinder the parallel solution of subproblems and the use of the efficient alternating directions method of multipliers. We show that this unfortunate situation is not the case for MDO problems with block-separable linking constraints. We derive a centralized formulation of ALC for block-separable constraints, which does allow parallel solution of subproblems. Similarly, we derive a distributed coordination variant for which subproblems cannot be solved in parallel, but that still enables the use of the alternating direction method of multipliers. The approach can also be used for other existing MDO coordination strategies such that they can include block-separable linking constraints. This work is funded by MicroNed, grant number 10005898.     

A solution process for simulation-based multiobjective design optimization with an application in the paper industry

In this paper, we address some computational challenges arising in complex simulation-based design optimization problems. High computational cost, black-box formulation and stochasticity are some of the challenges related to optimization of design problems involving the simulation of complex mathematical models. Solving becomes even more challenging in case of multiple conflicting objectives that must be optimized simultaneously. In such cases, application of multiobjective optimization methods is necessary in order to gain an understanding of which design offers the best possible trade-off. We apply a three-stage solution process to meet the challenges mentioned above. As our case study, we consider the integrated design and control problem in paper mill design where the aim is to decrease the investment cost and enhance the quality of paper on the design level and, at the same time, guarantee the smooth performance of the production system on the operational level. In the first stage of the three-stage solution process, a set of solutions involving different trade-offs is generated with a method suited for computationally expensive multiobjective optimization problems using parallel computing. Then, based on the generated solutions an approximation method is applied to create a computationally inexpensive surrogate problem for the design problem and the surrogate problem is solved in the second stage with an interactive multiobjective optimization method. This stage involves a decision maker and her/his preferences to find the most preferred solution to the surrogate problem. In the third stage, the solution best corresponding that of stage two is found for the original problem.     

基于微粒群算法的转炉合金加入量多目标优化模型

针对转炉炼钢过程中钢水出钢环节脱氧合金剂加入量确定的问题,建立了合金加入量优化计算模型.首先,利用支持向量机建立钢包元素含量预测模型,将该模型的预测结果与元素规定含量的误差作为优化模型的一个目标函数,合金成本作为另一个目标函数;然后利用预测收得率方法计算合金加入量并作为优化模型的初始值,使用改进的动态邻域多目标微粒群算法对优化模型进行求解,有效地找出Pareto最优解集,并根据实际需要得出较好的优化结果.仿真实验表明了该优化模型的有效性.