Multiobjective optimization using an adaptive weighting scheme

A new Pareto front approximation method is proposed for multiobjective optimization problems (MOPs) with bound constraints. The method employs a hybrid optimization approach using two derivative-free direct search techniques, and intends to solve black box simulation-based MOPs where the analytical form of the objectives is not known and/or the evaluation of the objective function(s) is very expensive. A new adaptive weighting scheme is proposed to convert a multiobjective optimization problem to a single objective optimization problem. Another contribution of this paper is the generalization of the star discrepancy-based performance measure for problems with more than two objectives. The method is evaluated using five test problems from the literature, and a realistic engineering problem. Results show that the method achieves an arbitrarily close approximation to the Pareto front with a good collection of well-distributed nondominated points for all six test problems.     

Multiobjective and multi-physics topology optimization using an updated smart normal constraint bi-directional evolutionary structural optimization method

To date the design of structures using topology optimization methods has mainly focused on single-objective problems. Since real-world design problems typically involve several different objectives, most of which counteract each other, it is desirable to present the designer with a set of Pareto optimal solutions that capture the trade-off between these objectives, known as a smart Pareto set. Thus far only the weighted sums and global criterion methods have been incorporated into topology optimization problems. Such methods are unable to produce evenly distributed smart Pareto sets. However, recently the smart normal constraint method has been shown to be capable of directly generating smart Pareto sets. Therefore, in the present work, an updated smart Normal Constraint Method is combined with a Bi-directional Evolutionary Structural Optimization (SNC-BESO) algorithm to produce smart Pareto sets for multiobjective topology optimization problems. Two examples are presented, showing that the Pareto solutions found by the SNC-BESO method make up a smart Pareto set. The first example, taken from the literature, shows the benefits of the SNC-BESO method. The second example is an industrial design problem for a micro fluidic mixer. Thus, the problem is multi-physics as well as multiobjective, highlighting the applicability of such methods to real-world problems. The results indicate that the method is capable of producing smart Pareto sets to industrial problems in an effective and efficient manner.     

Multiobjective optimization using population-based extremal optimization

In recent years, a general-purpose local-search heuristic method called Extremal Optimization (EO) has been successfully applied in some NP-hard combinatorial optimization problems. In this paper, we present a novel Pareto-based algorithm, which can be regarded as an extension of EO, to solve multiobjective optimization problems. The proposed method, called Multiobjective Population-based Extremal Optimization (MOPEO), is validated by using five benchmark functions and metrics taken from the standard literature on multiobjective evolutionary optimization. The experimental results demonstrate that MOPEO is competitive with the state-of-the-art multiobjective evolutionary algorithms. Thus MOPEO can be considered as a viable alternative to solve multiobjective optimization problems.     

A gradient-based path optimization method for motion planning

Most algorithms in probabilistic sampling-based path planning compute collision-free paths made of straight line segments lying in the configuration space. Due to the randomness of sampling, the paths make detours that need to be optimized. The contribution of this paper is to propose a basic gradient-based algorithm that transforms a polygonal collision-free path into a shorter one. While requiring only collision checking, and not any time-consuming obstacle distance computation nor geometry simplification, we constrain only part of the configuration variables that may cause a collision, and not entire configurations. Thus, parasite motions that are not useful for the problem resolution are reduced without any assumption. Experimental results include navigation and manipulation tasks, eg a manipulator arm-filling boxes and a PR2 robot working in a kitchen environment. Comparisons with a random shortcut optimizer and a partial shortcut have also been studied.     

A gradient-based transformation method in multidisciplinary design optimization

Multidisciplinary design optimization (MDO) has become essential for solving the complex engineering design problems. The most common approach is to “divide and conquer” the MDO problem, that is, to decompose the complex problem into several sub-problems and to collect the local solutions to give a new design point for the original problem. In 1990s, researchers have developed some decomposition strategies to find or synthesize the optimal model of the optimization structure in order to evenly distribute the computational workloads to multiple processors. Several MDO methods, such as Collaborative Optimization (CO) and Analytical Target Cascading (ATC), were then developed to solve the decomposed sub-problems and coordinate the coupling variables among them to find the optimal solution. However, both the synthesis of the decomposition structure and the coordination of the coupling variables require additional function evaluations, in terms of evaluating the functional dependency between each sub-problem and determining the proper weighting coefficients between each coupling functions respectively. In this paper, a new divide-and-conquer strategy, Gradient-based Transformation Method (GTM), is proposed to overcome the challenges in structure synthesis and variable coordination. The proposed method first decomposes the MDO problem into several sub-systems and distributes one constraint from the original problem to each sub-system without evaluating the dependency between each sub-system. Each sub-system is then transformed to the single-variate coordinate along the gradient direction of the constraint. The total function evaluations equal the number of constraints times the number of variables plus one in every iteration. Due to the monotonicity characteristics of the transformed sub-problems, they are efficiently solved by Monotonicity Analyses without any additional function evaluations. Two coordination principles are proposed to determine the significances of the responses based on the feasibility and activity conditions of every sub-problem and to find the new design point at the average point of the most significant responses. The coordination principles are capable of finding the optimal solution in the convex feasible space bounded by the linearized sub-system constraints without additional function evaluations. The optimization processes continue until the convergence criterion is satisfied. The numerical examples show that the proposed methodology is capable of effectively and efficiently finding the optimal solutions of MDO problems.     

Multiobjective optimization using an immunodominance and clonal selection inspired algorithm

Based on the mechanisms of immunodominance and clonal selection theory, we propose a new multiobjective optimization algorithm, immune dominance clonal multiobjective algorithm （IDCMA）. IDCMA is unique in that its fitness values of current dominated individuals are assigned as the values of a custom distance measure, termed as Ab-Ab affinity, between the dominated individuals and one of the nondominated individuals found so far. According to the values of Ab-Ab affinity, all dominated individuals （antibodies） are divided into two kinds, subdominant antibodies and cryptic antibodies. Moreover, local search only applies to the subdominant antibodies, while the cryptic antibodies are redundant and have no function during local search, but they can become subdominant （active） antibodies during the subsequent evolution. Furthermore, a new immune operation, clonal proliferation is provided to enhance local search. Using the clonal proliferation operation, IDCMA reproduces individuals and selects their improved maturated progenies after local search, so single individuals can exploit their surrounding space effectively and the newcomers yield a broader exploration of the search space. The performance comparison of IDCMA with MISA, NSGA-Ⅱ, SPEA, PAES, NSGA, VEGA, NPGA, and HLGA in solving six well-known multiobjective function optimization problems and nine multiobjective 0/1 knapsack problems shows that IDCMA has a good performance in converging to approximate Pareto-optimal fronts with a good distribution.     

Multiobjective structural optimization using a microgenetic algorithm

In this paper, we present a genetic algorithm with a very small population and a reinitialization process (a microgenetic algorithm) for solving multiobjective optimization problems. Our approach uses three forms of elitism, including an external memory (or secondary population) to keep the nondominated solutions found along the evolutionary process. We validate our proposal using several engineering optimization problems taken from the specialized literature and compare our results with respect to two other algorithms (NSGA-II and PAES) using three different metrics. Our results indicate that our approach is very efficient (computationally speaking) and performs very well in problems with different degrees of complexity.     

Multiobjective clustering analysis using particle swarm optimization

Clustering is a significant data mining task which partitions datasets based on similarities among data. This technique plays a very important role in the rapidly growing field known as exploratory data analysis. A key difficulty of effective clustering is to define proper grouping criteria that reflect fundamentally different aspects of a good clustering solution such as compactness and separation of clusters. Moreover, in the conventional clustering algorithms only a single criterion is considered that may not conform to the diverse and complex shapes of the underlying clusters. In this study, partitional clustering is defined as a multiobjective optimization problem. The aim is to obtain well-separated, connected, and compact clusters and for this purpose, two objective functions have been defined based on the concepts of data connectivity and cohesion. These functions are the core of an efficient multiobjective particle swarm optimization algorithm, which has been devised for and applied to automatic grouping of large unlabeled datasets. A comprehensive experimental study is conducted and the obtained results are compared with the results of four other state-of-the-art clustering techniques. It is shown that the proposed algorithm can achieve the optimal number of clusters, is robust and outperforms, in most cases, the other methods on the selected benchmark datasets.     

Multiobjective optimization of trusses using genetic algorithms

In this paper we propose the use of the genetic algorithm (GA) as a tool to solve multiobjective optimization problems in structures. Using the concept of min–max optimum, a new GA-based multiobjective optimization technique is proposed and two truss design problems are solved using it. The results produced by this new approach are compared to those produced by other mathematical programming techniques and GA-based approaches, proving that this technique generates better trade-offs and that the genetic algorithm can be used as a reliable numerical optimization tool.     

Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization

Gradient-based aerodynamic shape optimization using computational fluid dynamics (CFD), and time dependent problems in aeroelasticity, that is, coupled calculations between computational structural mechanics (CSM) and CFD, require repeated deformations of the CFD mesh.An interpolation scheme, based on radial basis functions (RBF), is devised in order to propagate the deformations from the boundaries to the interior of the CFD mesh. This method can lower the computational costs due to the deformation of the mesh, in comparison with the usual Laplace smoothing. Moreover, the algorithm is independent of the mesh connectivities. Therefore, structured and unstructured meshes are equally treated as well as hybrid meshes.The application of this interpolation scheme in problems of aerodynamic shape optimization is also carefully investigated. When the optimization is executed by a gradient-based algorithm the cost function is differentiated with respect to the design parameters in order to obtain the gradient. The gradient is most efficiently and accurately calculated by solving a certain adjoint equation derived from the discretized flow equations. The calculation of the gradient, which is detailed in this presentation, involves the Jacobian matrix of the mesh deformation.Finally, we present the results of an optimization of the ONERA M6 wing at transonic speed using the interpolation algorithm. The results are used for comparison with another technique of mesh deformation. The quality of the mesh obtained by the new algorithm, and the interpolation error, are analyzed with respect to the parameters of the interpolation scheme: the type of RBF, the RBF’s shape parameter, and the sets of control points.     

Pocketing toolpath computation using an optimization method

This paper proposes a new method of pocketing toolpath computation based on an optimization problem with constraints. Generally, the calculated toolpath has to minimize the machining time and respect a maximal effort on the tool during machining. Using this point of view, the toolpath can be considered as the result of an optimization in which the objective is to minimize the travel time and the constraints are to check the forces applied to the tool. Thus a method based on this account and using an optimization algorithm is proposed to compute toolpaths for pocket milling. After a review of pocketing toolpath computation methods, the framework of the optimization problem is defined. A modeling of the problem is then proposed and a solving method is presented. Finally, applications and experiments on machine tools are studied to illustrate the advantages of this method.     

Constraint handling in genetic algorithms using a gradient-based repair method

Constraint handling is one of the major concerns when applying genetic algorithms (GAs) to solve constrained optimization problems. This paper proposes to use the gradient information derived from the constraint set to systematically repair infeasible solutions. The proposed repair procedure is embedded into a simple GA as a special operator. Experiments using 11 benchmark problems are presented and compared with the best known solutions reported in the literature. Our results are competitive, if not better, compared to the results reported using the homomorphous mapping method, the stochastic ranking method, and the self-adaptive fitness formulation method.     

A simple method for coupled acoustic-mechanical analysis with application to gradient-based topology optimization

Structural and Multidisciplinary Optimization - A simple computational framework for analysis of acoustic-mechanical coupling is proposed. The method is based on extended finite element models for...     

Multiobjective optimization using variable complexity modelling for control system design

A multi-stage design approach that uses a multiobjective genetic algorithm as the framework for optimization and multiobjective preference articulation, and an H_infty loop-shaping technique are used to design controllers for a gas turbine engine. A non-linear model is used to assess performance of the controller. Because the computational load of applying multiobjective genetic algorithm to this control strategy is very high, a neural network and response surface models are used in order to speed up the design process within the framework of a multiobjective genetic algorithm. The final designs are checked using the original non-linear model.     

Multiobjective optimization of complex antenna structures using response surface models

A very efficient multiobjective (MO) design technique for complex antenna structures involving a large number of design parameters is presented. This design technique, multiobjective‐fractional factorial design (MO‐FFD), is very different from conventional Pareto‐based MO algorithms, which take a great deal of effort to balance the trade‐off between all the design specifications. By performing one single combination of simulations, all the response surface models of design goals are simultaneously built, and Derringer's desirability functions are readily applied to these models so that the optimum structure is obtained. Compared to classical MO algorithms such as Strength Pareto Evolutionary Algorithm 2, nondominated sorting particle swarm optimizer, and cultural MO particle swarm optimization, MO‐FFD yields more desirable performances yet the required number of simulations is reduced by 97%. This article thoroughly illustrates the mathematical development of MO‐FFD, deriving a novel application of ultrawideband (UWB) antennas because of its MO optimization capability. More explicitly, MO‐FFD overcomes all the design challenges of dual band‐notched UWB antennas including desired impedance characteristics, enhanced fidelity factors, and uniform peak gains over the passband, which are what conventional Pareto‐based algorithms cannot attain. The measured results show that all the performance criteria are met; especially, the time‐domain signal distortion is minimized. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:62–71, 2016.     

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.     

Optical flow estimation: an error analysis of gradient-based methods with local optimization

Multiple views of a scene can provide important information about the structure and dynamic behavior of three-dimensional objects. Many of the methods that recover this information require the determination of optical flow-the velocity, on the image, of visible points on object surfaces. An important class of techniques for estimating optical flow depend on the relationship between the gradients of image brightness. While gradient-based methods have been widely studied, little attention has been paid to accuracy and reliability of the approach. Gradient-based methods are sensitive to conditions commonly encountered in real imagery. Highly textured surfaces, large areas of constant brightness, motion boundaries, and depth discontinuities can all be troublesome for gradient-based methods. Fortunately, these problematic areas are usually localized can be identified in the image. In this paper we examine the sources of errors for gradient-based techniques that locally solve for optical flow. These methods assume that optical flow is constant in a small neighborhood. The consequence of violating in this assumption is examined. The causes of measurement errors and the determinants of the conditioning of the solution system are also considered. By understanding how errors arise, we are able to define the inherent limitations of the technique, obtain estimates of the accuracy of computed values, enhance the performance of the technique, and demonstrate the informative value of some types of error.     

A gradient-based,parameter-free approach to shape optimization

In shape optimization, the independent node movement approach, wherein finite element node coordinates are used directly as design variables, allows the most freedom for shape change and avoids the time-consuming parameterization preprocess. However, this approach lacks a length scale control that is necessary to ensure a well-posed shape optimization problem and avoid numerical instability. Motivated by the success of filtering techniques that impose minimum length scales in topology optimization, we propose a scheme with consistent filtering to introduce a length scale and thereby ensure smoothness in shape optimization while preserving the advantages of the independent node movement approach.     

Multiobjective topology optimization of structures using genetic algorithms with chromosome repairing

In this work, a genetic algorithm (GA) for multiobjective topology optimization of linear elastic structures is developed. Its purpose is to evolve an evenly distributed group of solutions to determine the optimum Pareto set for a given problem. The GA determines a set of solutions to be sorted by its domination properties and a filter is defined to retain the Pareto solutions. As an equality constraint on volume has to be enforced, all chromosomes used in the genetic GA must generate individuals with the same volume value; in the coding adopted, this means that they must preserve the same number of “ones” and, implicitly, the same number of “zeros” along the evolutionary process. It is thus necessary: (1) to define chromosomes satisfying this propriety and (2) to create corresponding crossover and mutation operators which preserve volume. Optimal solutions of each of the single-objective problems are introduced in the initial population to reduce computational effort and a repairing mechanism is developed to increase the number of admissible structures in the populations. Also, as the work of the external loads can be calculated independently for each individual, parallel processing was used in its evaluation. Numerical applications involving two and three objective functions in 2D and two objective functions in 3D are employed as tests for the computational model developed. Moreover, results obtained with and without chromosome repairing are compared.     

连续过程的多目标优化

在模块环境(Aspen Plus)下,建立了基于多目标遗传算法NSGA-Ⅱ求解多目标优化问题的系统结构,并对含循环物流的连续过程废料最小化问题进行求解。在求解过程中遗传算法需要反复调用流程模拟,而流程中循环物流的迭代收敛使优化计算效率较低。为减少流程迭代次数本文提出2个加速策略:一是变收敛精度策略,在优化计算初始阶段,使流程在较低精度下收敛,快速淘汰劣点,随着优化的进行,将流程收敛精度逐步提高,得到高质量的非劣解;二是循环流初值策略,利用已有的计算值,回归决策变量与循环流变量的对应关系,改善循环流初值。实例结果表明,加速策略减少了一半左右的流程迭代次数,效率提高50%,本文提出的求解多目标问题的方法能方便地得到问题的Pareto最优解集,可应用于一般连续化工过程的多目标优化。