An efficient multi-objective optimization method for black-box functions using sequential approximate technique

Multi-objective optimization problems in practical engineering usually involve expensive black-box functions. How to reduce the number of function evaluations at a good approximation of Pareto frontier has been a crucial issue. To this aim, an efficient multi-objective optimization method based on a sequential approximate technique is suggested in this paper. In each iteration, according to the prediction of radial basis function with a micro multi-objective genetic algorithm, an extended trust region updating strategy is adopted to adjust the design region, a sample inheriting strategy is presented to reduce the number of new function evaluations, and then a local-densifying strategy is proposed to improve the accuracy of approximations in concerned regions. At the end of each iteration, the obtained actual Pareto optimal points are stored in an external archive and are updated as the iteration process. The effect of the present method is demonstrated by eight test functions. Finally, it is employed to perform the structure optimization of a vehicle door.     

Shape optimization of heavy load carrying components for structural performance and manufacturing cost

This paper addresses the trade-off between structural performance and manufacturing cost of heavy load carrying components by incorporating virtual machining (VM) technique in computer-aided design (CAD)-based shape optimization problem. A structural shape optimization problem is set up to minimize total cost, subject to the limits on structural performance measures. For every design iteration, finite element analysis (FEA) is conducted to evaluate structural performance, and VM is employed to ascertain machinability and estimate machining time. Design sensitivity coefficients of objective function and constraints are computed and supplied to the optimization algorithm. Based on the gradients, the algorithm determines design changes, which are used to update FEA and VM models. The process is repeated until specified convergence criterion is satisfied. Application programs developed to integrate commercially available CAD/CAM/FEA/Design optimization tools enable implementation in virtual environment and facilitate automation. The application programs can be reused for similar design problems provided that the same set of tools is used.     

A sequential approximate programming strategy for reliability-based structural optimization

Although reliability-based structural optimization (RBSO) is recognized as a rational structural design philosophy that is more advantageous to deterministic optimization, most common RBSO is based on straightforward two-level approach connecting algorithms of reliability calculation and that of design optimization. This is achieved usually with an outer loop for optimization of design variables and an inner loop for reliability analysis. A number of algorithms have been proposed to reduce the computational cost of such optimizations, such as performance measure approach, semi-infinite programming, and mono-level approach. Herein the sequential approximate programming approach, which is well known in structural optimization, is extended as an efficient methodology to solve RBSO problems. In this approach, the optimum design is obtained by solving a sequence of sub-programming problems that usually consist of an approximate objective function subjected to a set of approximate constraint functions. In each sub-programming, rather than direct Taylor expansion of reliability constraints, a new formulation is introduced for approximate reliability constraints at the current design point and its linearization. The approximate reliability index and its sensitivity are obtained from a recurrence formula based on the optimality conditions for the most probable failure point (MPP). It is shown that the approximate MPP, a key component of RBSO problems, is concurrently improved during each sub-programming solution step. Through analytical models and comparative studies over complex examples, it is illustrated that our approach is efficient and that a linearized reliability index is a good approximation of the accurate reliability index. These unique features and the concurrent convergence of design optimization and reliability calculation are demonstrated with several numerical examples.     

改进的量子粒子群多目标优化算法

针对粒子群优化算法容易陷入局部极值点的问题,提出了一种新的量子比特粒子群算法,该算法采用Pareto支配关系来更新粒子的个体最优值和局部最优值;定义极大极小距离,并采用该距离方法裁减非支配解.实验结果表明该算法能更好地接近Pareto前沿且具有更好的分布性,更适合于求解复杂高维优化问题,是一种非常有潜力的多目标优化方法.     

A survey of multi-objective metaheuristics applied to structural optimization

Structural and Multidisciplinary Optimization - In civil and industrial engineering, structural design optimization problems are usually characterized by the presence of multiple conflicting...     

一种新的多目标粒子群优化算法

论文提出了一种基于拥挤度和动态惯性权重聚合的多目标粒子群优化算法,该算法采用Pareto支配关系来更新粒子的个体最优值,用外部存档策略保存搜索过程中发现的非支配解;采用适应值拥挤度裁剪归档中的非支配解,并从归档中的稀松区域随机选取精英作为粒子的全局最优位置,以保持解的多样性;采用动态惯性权重聚合的方法以使算法尽可能地逼近各目标的最优解。仿真结果表明,该算法性能较好,能很好地求解多目标优化问题。     

A minimax reduction method for multi-objective decomposition-based design optimization

In this paper, a method for the design optimization of multi-objective engineering problems (or systems) which are decomposedhierarchically ornonhierarchically into several subproblems is presented. The method is based on a minimax formulation of the overall problem and the application of some reduction measures to reduce the number of variables in this problem. Two well-known examples are selected from the literature and included to demonstrate the proposed solution steps.     

A multi-objective method of hinge-free compliant mechanism optimization

A new multi-objective formulation for topology synthesis of hinge-free compliant mechanisms is presented based on the SIMP method. A weighted sum formed objective function is developed by taking into consideration the input and output mean compliances. The weighting factors are set based on the information that is obtained from the previous iteration and automatically updated with each optimization iteration step. Shape sensitivity analysis is addressed. Some numerical examples are presented to illustrate the validity of the proposed method.     

Integration of design and manufacturing for structural shape optimization

This paper presents an integrated design and manufacturing approach that supports shape optimization of structural components. The approach starts from a primitive concept stage, where boundary and loading conditions of the structural component are given to the designer. Topology optimization is conducted for an initial structural layout. The discretized structural layout is smoothed using parametric B-Spline surfaces. The B-Spline surfaces are imported into a CAD system to construct parametric solid models for shape optimization. Virtual manufacturing (VM) techniques are employed to ensure that the optimized shape can be manufactured at a reasonable cost. The solid freeform fabrication (SFF) system fabricates physical prototypes of the structure for design verification. Finally, a computer numerical control (CNC) machine is employed to fabricate functional parts as well as mold or die for mass production of the structural component. The main contribution of the paper is incorporating manufacturing into the design process, where manufacturing cost is considered for design. In addition, the overall design process starts from a primitive stage and ends with functional parts. A 3D tracked vehicle roadarm is employed throughout this paper to illustrate the overall design process and various techniques involved.     

A discrete post-processing method for structural optimization

This paper introduces a discrete variable post-processing method for structural design optimization. The motivation behind the method is to find a good discrete solution at manageable cost while the traditional discrete optimization algorithms are regarded as impractical for large-scale structural design problems. In this paper, the Design of Experiments (DOE) and Conservative Discrete Design (CDD) approaches have been proposed to deal with discrete variables with limited computational cost. Both methods work on the explicit approximate discreteproblem to explore the discrete design. These two approaches, together with engineering rounded-off methods, can be used to process discrete variables at any specified continuous design optimization cycle for structural design problems. Brief background and a theoretical discussion about these approaches are given in this paper. Finally, the methods that have been implemented in MSC.Nastran are demonstrated by academic and real engineering examples.     

A pseudo-discrete rounding method for structural optimization

A new heuristic method aimed at efficiently solving the mixed-discrete nonlinear programming (MDNLP) problem in structural optimization, and denotedselective dynamic rounding, is presented. The method is based on the sequential rounding of a continuous solution and is in its current form used for the optimal discrete sizing design of truss structures. A simple criterion based on discrete variable proximity is proposed for selecting the sequence in which variables are to be rounded, and allowance is made for both upward and downward rounding. While efficient in terms of the required number of function evaluations, the method is also effective in obtaining a low discrete approximation to the global optimum. Numerical results are presented to illustrate the effectiveness and efficiency of the method.     

一种云计算资源的多目标优化的调度方法

云计算通过虚拟化技术将基础设施硬件资源虚拟化,以动态可缩放的方式提供给用户。云计算基础设施规模不断增加导致资源调度系统负载不均衡,从而造成资源浪费等问题。提出多目标优化资源调度策略和相应的算法,试图同时满足多个资源调度优化目标,如减少资源浪费,降低服务等级约定(SLA)违背率、保持系统负载均衡等。通过仿真实验,验证了多目标优化资源调度的策略能够在多个相互冲突的目标之间实现最优权衡。     

Computational efficiency of improved move limit method of sequential linear programming for structural optimization

The imporved move limit method of sequential linear programming is briefly explained. Comparison of computing efficiencies is made between the improved method and the conventional move limit method with six test problems. The usefulness of the method in the context of structural optimization is shown with the help of four examples.     

A metaheuristic approach for selecting a common platform for modular products based on product performance and manufacturing cost

A known strategy to implement Mass Customization is Product Modularization. To take advantage of the benefits of modularity, the selection of a common platform is required. This selection must be done with optimization criteria based on functionality and economics. In this paper we propose metaheuristic procedures to solve the problem of selecting a common platform for a modular product. This selection is based on an aggregate objective function that combines product performance and manufacturing cost. The problem is divided into two hierarchical problems that must be solved sequentially. The mathematical models have a non-linear integer formulation. Because of the computational complexity to solve optimally these models, metaheuristic procedures are proposed to solve each sub-problem. These procedures are based on Scatter Search and Tabu Search. A case study is presented with a small instance that is solved with these procedures and by total enumeration. The results of the metaheuristic procedures coincide with the optimal values found by total enumeration. The run times are reasonable and it is expected a greater benefit for a larger instance with similar results quality.     

云制造环境下面向多目标优化的虚拟资源调度研究

为解决云制造环境下虚拟资源调度存在的算法求解效率不高、模型建立缺乏考虑任务间关系约束和任务间及子任务间的物流时间及成本因素等不足,构建了兼顾交货期时间最小化、服务成本最低化、服务质量最优化为目标的多目标虚拟资源调度模型;采用一种基于项目阶段的双链编码方式进行编码,并提出自适应交叉与变异概率公式,以避免交叉、变异概率始终不变导致算法效率下降与过早收敛的问题;在此基础上利用基于项目阶段的多种交叉变异策略相结合的改进遗传算法进行求解,保证了算法的全局与局部搜索性能。实例结果表明,相比于传统的模型与算法,该模型适用性更强,改进的遗传算法在求解效率、准确度与稳定性方面均有较大提高。     

A semi-Lagrangian level set method for structural optimization

In level set based structural optimization, semi-Lagrange method has an advantage to allow for a large time step without the limitation of Courant–Friedrichs–Lewy (CFL) condition for numerical stability. In this paper, a line search algorithm and a sensitivity modulation scheme are introduced for the semi-Lagrange method. The line search attempts to adaptively determine an appropriate time step in each iteration of optimization. With consideration of some practical characteristics of the topology optimization process, incorporating the line search into semi-Lagrange optimization method can yield fewer design iterations and thus improve the overall computational efficiency. The sensitivity modulation is inspired from the conjugate gradient method in finite-dimensions, and provides an alternative to the standard steepest descent search in level set based optimization. Two benchmark examples are presented to compare the sensitivity modulation and the steepest descent techniques with and without the line search respectively.     

A stochastic method for discrete and continuous optimization in manufacturing systems

Algorithms for discrete and continuous optimization are a very important part of decision making systems in manufacturing. Most planning, scheduling and layout problems require these algorithms. In practice, research into efficient algorithms meets two principal obstacles. The first one is linked to the fact that quite often the criteria cannot be expressed in an analytic form, so it is not possible to use existing theoretical resolution methods. The second is due to the fact that most of the problems for which the criteria can be represented in analytic form are NP-hard problems. This situation can be simplified using simulation. But the use of simulation and optimization methods together often gives a local optimum. The proposed method in this paper is based on the use of a discrete modification of -transforms jointly with some heuristics for local optimization. The originality of this approach is in the possibility to avoid a local optimum, while using models of simulation for the computation of values of the criteria. An example of the utilization of the method is given: it concerns the optimization of the launching of the parts in production in systems of the job-shop type. The proposed method is compared with a heuristic known to be very good in the same number of simulations. The results of five tests with different model sizes show the efficiency of the proposed method.     

A modified hypervolume based expected improvement for multi-objective efficient global optimization method

The hypervolume indicator has been proved as an outstanding metric for the distribution of Pareto points, and the derived hypervolume based expected improvement (HVEI) has received a particular attention in the multi-objective efficient global optimization (EGO) method. However, the high computational cost has become the bottle neck which limits the application of HVEI on many objective optimization. Aiming at this problem, a modified version of HVEI (MHVEI) is proposed in this paper, which is easier to implement, maintains all the desired properties, and has a much lower computational cost. The theoretical study shows that the new criterion can be considered as a weighted integral form of HVEI, and it prefers the new point with a higher uncertainty compared with HVEI. The numerical tests show that the MHVEI performs similar as HVEI on the lower dimensional problem, and the advantage of MHVEI becomes more obvious as the dimension grows.