Uncertainty quantifications of Pareto optima in multiobjective problems

Design is a multi-objective decision-making process considering manufacturing, cost, aesthetics, usability among many other product attributes. The set of optimal solutions, the Pareto set, indicates the trade-offs between objectives. Decision-makers generally select their own optima from the Pareto set based on personal preferences or other judgements. However, uncertainties from manufacturing processes and from operating conditions will change the performances of the Pareto optima. Evaluating the impacts of uncertainties on Pareto optima requires a large amount of data and resources. Comparing multiple Pareto solutions under uncertainty are also very costly. In this work, local Pareto set approximation is integrated with uncertainty propagation technique to quantify design variations in the objective space. An optimality influence range is proposed using linear combinations of objective functions that creates a more accurate polygon objective variation subspace. A set of ‘virtual samples’ is then generated to form two quantifications of the objective variation subspace, namely an influence noise to indicate how a design remains optimal, and an influence range that quantifies the overall variations of a design. In most engineering practices, a Pareto optimum with a smaller influence noise and a smaller influence range is preferred. We also extend the influence noise/range concept to nonlinear Pareto set with the second-order approximation. The quadratic local Pareto approximation method in the literature is also extended in this work to solve multi-objective engineering problems with black-box functions. The usefulness of the proposed quantification method is demonstrated using a numerical example as well as using an engineering problem in structural design.     

A hybrid variable neighborhood search for solving the hybrid flow shop scheduling problem

This paper proposes a hybrid variable neighborhood search (HVNS) algorithm that combines the chemical-reaction optimization (CRO) and the estimation of distribution (EDA), for solving the hybrid flow shop (HFS) scheduling problems. The objective is to minimize the maximum completion time. In the proposed algorithm, a well-designed decoding mechanism is presented to schedule jobs with more flexibility. Meanwhile, considering the problem structure, eight neighborhood structures are developed. A kinetic energy sensitive neighborhood change approach is proposed to extract global information and avoid being stuck at the local optima. In addition, contrary to the fixed neighborhood set in traditional VNS, a dynamic neighborhood set update mechanism is utilized to exploit the potential search space. Finally, for the population of local optima solutions, an effective EDA-based global search approach is investigated to direct the search process to promising regions. The proposed algorithm is tested on sets of well-known benchmark instances. Through the analysis of experimental results, the high performance of the proposed HVNS algorithm is shown in comparison with four efficient algorithms from the literature.     

基于均匀设计的聚类多目标粒子群优化算法

为更有效地求解多目标优化问题,提出一种基于均匀设计的聚类多目标粒子群算法UCMOPSO。采用基于均匀设计的交叉操作尽可能地获得目标空间中均匀分布的非劣解,帮助种群跳出局部最优解,并通过一种新的聚类操作选择外部存档中有代表性的非劣解,从而控制外部存档规模,降低计算复杂度。对基准函数的测试结果表明,UCMOPSO算法相比同类算法在收敛性和分布性方面具有优势。     

Optimal synthesis of planar mechanisms via an extensible-link approach

This paper presents a novel approach to optimize the design of planar mechanisms with revolute joints for function generation or path synthesis. The proposed method is based on the use of an extensible-link mechanism model whose strain energy is minimized to find the optimal rigid design. This enables us to get rid of assembly constraints and the use of natural coordinates makes the objective function simpler. Two optimization strategies are developed and then discussed. The first one relies on alternate optimizations of design parameters and point coordinates. The second one uses multiple partial syntheses as starting point for a full synthesis process. The question of finding the global optimum is also addressed and developed. A simple algorithm is proposed to find several local optima among which the designer may choose the best one taking other criteria into account (e.g. stiffness, collision, size,...). Three applications are presented to illustrate the strategies while mentioning their limits.     

A general approach forcing convexity of ply angle optimization in composite laminates

This paper deals with optimization of laminated composite structures in which the ply angles are taken as design variables. One of the major problems when using ply-angles as design variables, is the lack of convexity of the objective function and thus the existence of local optima, which implies that usual gradient based optimization procedures may not be effective. Therefore, a new general approach that avoids the abovementioned problems of nonconvexity when ply-angles are used as design variables is proposed. The methodology is based upon the fact that the design space for an optimization problem formulated in lamination parameters [introduced by Tsai and Pagano (1968)] is proven to be convex, because the laminate stiffnesses are expressed linearly in terms of the lamination parameters. However, lamination parameters have at least two major shortcomings: as yet, for the general case involving membrane-bending coupling, the constraints between the lamination parameters are not completely defined; also, for a prescribed set of lamination parameters physically realizable composite laminates (e.g. laminates with equal thickness plies) may not exist. The approach here, uses both lamination parameters and ply-angles and thereby uses the advantages of both and eliminates the shortcomings of both.In order to illustrate this approach, several stiffness optimization examples are provided.     

Max-min surrogate-assisted evolutionary algorithm for robust design

Solving design optimization problems using evolutionary algorithms has always been perceived as finding the optimal solution over the entire search space. However, the global optima may not always be the most desirable solution in many real-world engineering design problems. In practice, if the global optimal solution is very sensitive to uncertainties, for example, small changes in design variables or operating conditions, then it may not be appropriate to use this highly sensitive solution. In this paper, we focus on combining evolutionary algorithms with function approximation techniques for robust design. In particular, we investigate the application of robust genetic algorithms to problems with high dimensions. Subsequently, we present a novel evolutionary algorithm based on the combination of a max-min optimization strategy with a Baldwinian trust-region framework employing local surrogate models for reducing the computational cost associated with robust design problems. Empirical results are presented for synthetic test functions and aerodynamic shape design problems to demonstrate that the proposed algorithm converges to robust optimum designs on a limited computational budget.     

Particle swarm optimizer for variable weighting in clustering high-dimensional data

In this paper, we present a particle swarm optimizer (PSO) to solve the variable weighting problem in projected clustering of high-dimensional data. Many subspace clustering algorithms fail to yield good cluster quality because they do not employ an efficient search strategy. In this paper, we are interested in soft projected clustering. We design a suitable k-means objective weighting function, in which a change of variable weights is exponentially reflected. We also transform the original constrained variable weighting problem into a problem with bound constraints, using a normalized representation of variable weights, and we utilize a particle swarm optimizer to minimize the objective function in order to search for global optima to the variable weighting problem in clustering. Our experimental results on both synthetic and real data show that the proposed algorithm greatly improves cluster quality. In addition, the results of the new algorithm are much less dependent on the initial cluster centroids. In an application to text clustering, we show that the algorithm can be easily adapted to other similarity measures, such as the extended Jaccard coefficient for text data, and can be very effective.     

Self-adaptive mutations may lead to premature convergence

Self-adaptive mutations are known to endow evolutionary algorithms (EA) with the ability of locating local optima quickly and accurately, whereas it was unknown whether these local optima are finally global optima provided that the EA runs long enough. In order to answer this question, it is assumed that the (1+1)-EA with self-adaptation is located in the vicinity P of a local solution with objective function value ε. In order to exhibit convergence to the global optimum with probability one, the EA must generate an offspring that is an element of the lower level set S containing all solutions (including a global one) with objective function value less than ε. In case of multimodal objective functions, these sets P and S are generally not adjacent, i.e., min{∥x-y∥:x∈P, y∈S}>0, so that the EA has to surmount the barrier of solutions with objective function values larger than ε by a lucky mutation. It will be proven that the probability of this event is less than one even under an infinite time horizon. This result implies that the EA can get stuck at a nonglobal optimum with positive probability. Some ideas of how to avoid this problem are discussed as well     

Multiobjective satisfaction within an interactive evolutionary design environment

The paper introduces the concept of an Interactive Evolutionary Design System (IEDS) that supports the engineering designer during the conceptual/preliminary stages of the design process. Requirement during these early stages relates primarily to design search and exploration across a poorly defined space as the designer's knowledge base concerning the problem area develops. Multiobjective satisfaction plays a major role, and objectives are likely to be ill-defined and their relative importance uncertain. Interactive evolutionary search and exploration provides information to the design team that contributes directly to their overall understanding of the problem domain in terms of relevant objectives, constraints, and variable ranges. This paper describes the development of certain elements within an interactive evolutionary conceptual design environment that allows off-line processing of such information leading to a redefinition of the design space. Such redefinition may refer to the inclusion or removal of objectives, changes concerning their relative importance, or the reduction of variable ranges as a better understanding of objective sensitivity is established. The emphasis, therefore, moves from a multiobjective optimization over a preset number of generations to a relatively continuous interactive evolutionary search that results in the optimal definition of both the variable and objective space relating to the design problem at hand. The paper describes those elements of the IEDS relating to such multiobjective information gathering and subsequent design space redefinition.     

A radial space division based evolutionary algorithm for many-objective optimization

In evolutionary many-objective optimization, diversity maintenance plays an important role in pushing the population towards the Pareto optimal front. Existing many-objective evolutionary algorithms mainly focus on convergence enhancement, but pay less attention to diversity enhancement, which may fail to obtain uniformly distributed solutions or fall into local optima. This paper proposes a radial space division based evolutionary algorithm for many-objective optimization, where the solutions in high-dimensional objective space are projected into the grid divided 2-dimensional radial space for diversity maintenance and convergence enhancement. Specifically, the diversity of the population is emphasized by selecting solutions from different grids, where an adaptive penalty based approach is proposed to select a better converged solution from the grid with multiple solutions for convergence enhancement. The proposed algorithm is compared with five state-of-the-art many-objective evolutionary algorithms on a variety of benchmark test problems. Experimental results demonstrate the competitiveness of the proposed algorithm in terms of both convergence enhancement and diversity maintenance.     

Evolutionary multiobjective industrial design: the case of a racing car tire-suspension system

When dealing with multiobjective optimization (MO) of the tire-suspension system of a racing car, a large number of design variables and a large number of objectives have to be taken into account. Two different models have been used, both validated on data coming from an instrumented car, a differential equation-based physical model, and a neural network purely numerical model. Up to 23 objective functions have been defined, at least 14 of which are in strict conflict of each other. The equivalent scalar function based and the objective-as-constraint formulations are intentionally avoided due to their well-known limitations. A fuzzy definition of optima, being a generalization of Pareto optimality, is applied to the problem. The result of such an approach is that subsets of Pareto optimal solutions (on such a problem, a big portion of the entire search space) can be properly selected as a consequence of input from the designer. The obtained optimal solutions are compared with the reference vehicle and with the optima previously obtained with design of experiment techniques and different MO optimization strategies. The proposed strategy improves both the reference (actual) car and previously obtained optima (scalar preference function) in the majority of objectives with technically significant improvements. Moreover, the strategy offers an univoque criterion for the choice among tradeoff solutions in the 14-dimensional objective space. The problem is used as a test of a proposed optimal design strategy for industrial problems, integrating differential equation and neural networks modeling, design of experiments, MO, and fuzzy optimal-based decision making. Such a linked approach gives also a unified view of where to concentrate the computational effort.     

基于QPSO的纹理合成算法

针对PSO算法搜索空间有限,容易陷入局部最优点的缺陷,提出一种以块算法为基础,量子粒子群优化算法（QPSO）为优化策略的纹理合成方法。实验结果表明,与标准PSO算法相比,由于量子粒子群优化算法（QPSO）显著的全局收敛性,这种新型的纹理合成方法,使最后的合成图像中采样块结合处更流畅,纹理更细腻。     

Interval analysis based robust truss optimization with continuous and discrete variables using mix-coded genetic algorithm

The problem of optimizing truss structures in the presence of uncertain parameters considering both continuous and discrete design variables is studied. An interval analysis based robust optimization method combined with the improved genetic algorithm is proposed for solving the problem. Uncertain parameters are assumed to be bounded in specified intervals. The natural interval extensions are employed to obtain explicitly a conservative approximation of the upper and lower bounds of the structural response, and hereby the bounds of the objective function and the constraint function. This way the uncertainty design may be performed in a very efficient manner in comparison with the probabilistic analysis based method. A mix-coded genetic algorithm (GA), where the discrete variables are coded with binary numbers while the continuous variables are coded with real numbers, is developed to deal with simultaneously the continuous and discrete design variables of the optimization model. An improved differences control strategy is proposed to avoid the GA getting stuck in local optima. Several numerical examples concerning the optimization of plane and space truss structures with continuous, discrete or mixed design variables are presented to validate the method developed in the present paper. Monte Carlo simulation shows that the interval analysis based optimization method gives much more robust designs in comparison with the deterministic optimization method.     

Optimum weight design of sandwich cylinders with orthotropic facings and core under combined loads

The minimum weight of a honeycomb sandwich cylinder with the facings in composite material is obtained by an optimization method with the ply angles and the thicknesses of the ply and honeycomb as the design variables. The cylinder is under the combined loads of axial compression, bending moment and transverse shear. The structural analyses verify the stability and the material strength. The stability analyses are done with the simply supported boundary conditions for the orthotropic case. Three stability modes have been studied: general buckling, dimpling and wrinkling. In all cases, the initial imperfections and the prebuckling distortions are not taken into account. This problem is treated with the knock-down factors. For the optimization, we have used a variable metric method for constrained optimization (subroutine VMCON, method of M.J.D. Powell). From an arbitrary design point (feasible or non-feasible), the program searches for an optimal design by the iterative procedure. To verify that the obtained optima are not local, the program has been initiated from several different starting points. Having compared the results, global alternative optima as well as local optima have been found.     

Structural synthesis of frame reinforced,submersible, circular,cylindrical hulls

This paper describes a mathematical programming procedure for the automated optimal structural synthesis of frame stiffened, cylindrical shells. For a specified set of design parameters such as external pressure, shell radius and length and material properties, the method generates those values of the design variables that produce a minimum weight design. The skin, frame web and frame flange thicknesses and the flange width are treated as continuous variables. Frame spacing is considered a discrete variable. Constraint equations control local and general shell and frame instability and yield. Limits may be placed on the variable values, and certain geometric or space constraints can be applied. The mixed (continuous and discrete nonlinear programming problem is solved by a combination of a discrete ‘Golden Search’ for the optimal number of frames and the ‘Direct Search Design Algorithm’ which provides the optimum values of the continuous variables.     

Nonconvex and discrete optimal design using multiple state simulated annealing

New strategies used in multiple state simulated annealing are proposed with the goal of increasing the chances of locating more optima through the use of interactive search strategies. A multiple state simulated annealing is characterized as one in which multiple sequences of state changes, instead of only one, are independently created under a common temperature dropping schedule and state change process. A number of interactive strategies are proposed to interconnect the development of multiple states during the annealing process so that in a single run of miltiple state simulated annealing the design space could be explored more thoroughly and more global/local optima could be discovered. Two illustrative examples including nonconvex and discrete optimization problems are included.     

Hybrid meta-model based search method for expensive problems

The complexity and opaque characteristics of the practical expensive problems hinder the further applications of the single meta-model based optimization algorithms. In this work, a hybrid meta-model based search method (HMBSM) is presented. In this method, an important region is firstly constructed using a part of the expensive points which are evaluated by the expensive problems. Then, three meta-models with different fitting techniques are used together both in the important region and the remaining region. The whole design space will also be searched simultaneously to further avoid the local optima. Through intensive test by six benchmark math functions with the variables ranging from 10 to 24 and compared with the efficient global optimization (EGO), hybrid meta-model based design space management (HMDSM) method and multiple meta-model based design space differentiation (MDSD) method, the proposed HMBSM method shows excellent accuracy, efficiency and robustness. Then, the proposed method is applied in a vehicle lightweight design involving finite element analysis with 30 design variables, reducing 11.4 kg of weight.     

A multi-population cultural algorithm with adaptive diversity preservation and its application in ammonia synthesis process

A multi-population cultural differential evolution (MCDE) algorithm is proposed. Each of the populations is managed by its private cultural differential evolution algorithm, in which a center individual is introduced into the belief space and selection function follows a new method to select the offspring for the next generation. To accelerate the convergence speed, the populations exchange their knowledge with each other every given generations. An adaptive mechanism of population diversity preservation is put forward to prevent the populations from being trapped in local optima. In the adaptive mechanism, the idea of culture fusion between populations is used to know the convergence status, so that the diversity of populations is kept along the evolutionary process. The performance evaluation on MCDE using eleven constrained optimization problems shows that MCDE is a competitive approach. MCDE is further applied to a practical optimization problem in an ammonia synthesis system with the objective to maximize the net value of ammonia. The results achieved by MCDE are compared with those by two traditional differential evolution algorithms, which indicate that MCDE has more excellent performance and better effectiveness.     

Vaccine-Enhanced Artificial Immune System for Multimodal Function Optimization

This paper emulates a biological notion in vaccines to promote exploration in the search space for solving multimodal function optimization problems using artificial immune systems (AISs). In this method, we first divide the decision space into equal subspaces. The vaccine is then randomly extracted from each subspace. A few of these vaccines, in the form of weakened antigens, are then injected into the algorithm to enhance the exploration of global and local optima. The goal of this process is to lead the antibodies to unexplored areas. Using this biologically motivated notion, we design the vaccine-enhanced AIS for multimodal function optimization, achieving promising performance.      

A global algorithm for minimax solutions to a stochastic programming problem

A minimax solution to a stochastic program occurs when the objective function is maximized subject to the random parameters jointly taking on their most adverse or pessimistic values. Minimax solutions have been proposed for decision making in agricultural planning, and to provide a lower bound to the values of the objective function of the “wait and see” stochastic program. In this paper a non-convex minimax problem and the occurrence of local optima are discussed. A global algorithm is presented for the minimax problem of a stochastic program in which some of the right hand side parameters are stochastic. It is also shown how minimax solutions may be obtained where stochastic parameters occur solely in the objective function, and in the objective function and right hand sides simultaneously.