Pareto optimization using the struggle genetic crowding algorithm

Many real-world engineering design problems involve the simultaneous optimization of several conflicting objectives. In this paper, a method combining the struggle genetic crowding algorithm with Pareto-based population ranking is proposed to elicit trade-off frontiers. The new method has been tested on a variety of published problems, reliably locating both discontinuous Pareto frontiers as well as multiple Pareto frontiers in multi-modal search spaces. Other published multi-objective genetic algorithms are less robust in locating both global and local Pareto frontiers in a single optimization. For example, in a multi-modal test problem a previously published non-dominated sorting GA (NSGA) located the global Pareto frontier in 41% of the optimizations, while the proposed method located both global and local frontiers in all test runs. Additionally, the algorithm requires little problem specific tuning of parameters.     

Formulation and application of weight-function-based physical programming

Physical programming is effective in multi-objective optimization since it assists the designer to find the most preferred solution. Preference-function-based physical programming (PFPP) abandons the weighted-sum approach and its performance in generating Pareto solutions is susceptible to the transformation of pseudo-preferences. With the aim of integrating a weighted-sum approach into physical programming and generating well-distributed Pareto solutions, a weight-function-based physical programming (WFPP) method has been proposed. The approach forms a weight function for each normalized criterion and uses the variable weighted sum of all criteria as the aggregate objective function. Implementation for numerical and engineering design problems indicates that WFPP works as well as PFPP. The design process of generating Pareto solutions by WFPP is further presented, where the pseudo-preferences are allowed to transform in different ranges. Examples and results demonstrate that solutions generated by WFPP have better diversity performance than those of PFPP, especially when the pseudo-preferences are far from the true Pareto front.     

Efficient Pareto Frontier Exploration using Surrogate Approximations

In this paper we present an efficient and effective method of using surrogate approximations to explore the design space and capture the Pareto frontier during multiobjective optimization. The method employs design of experiments and metamodeling techniques (e.g., response surfaces and kriging models) to sample the design space, construct global approximations from the sample data, and quickly explore the design space to obtain the Pareto frontier without specifying weights for the objectives or using any optimization. To demonstrate the method, two mathematical example problems are presented. The results indicate that the proposed method is effective at capturing convex and concave Pareto frontiers even when discontinuities are present. After validating the method on the two mathematical examples, a design application involving the multiobjective optimization of a piezoelectric bimorph grasper is presented. The method facilitates multiobjective optimization by enabling us to efficiently and effectively obtain the Pareto frontier and identify candidate designs for the given design requirements.     

Pareto Frontier Based Concept Selection Under Uncertainty,with Visualization

In a recent publication, we presented a new multiobjective decision-making tool for use in conceptual engineering design. In the present paper, we provide important developments that support the next phase in the evolution of the tool. These developments, together with those of our previous work, provide a concept selection approach that capitalizes on the benefits of computational optimization. Specifically, the new approach uses the efficiency and effectiveness of optimization to rapidly compare numerous designs, and characterize the tradeoff properties within the multiobjective design space. As such, the new approach differs significantly from traditional (non-optimization based) concept selection approaches where, comparatively speaking, significant time is often spent evaluating only a few points in the design space. Under the new approach, design concepts are evaluated using a so-calleds-Pareto frontier; this frontier originates from the Pareto frontiers of various concepts, and is the Pareto frontier for thesetof design concepts. An important characteristic of the s-Pareto frontier is that it provides a foundation for analyzing tradeoffs between design objectives and the tradeoffs between design concepts. The new developments presented in this paper include; (i) the notion ofminimally representingthe s-Pareto frontier, (ii) the quantification of concept goodness using s-Pareto frontiers, (iii) the development of an interactive design space exploration approach that can be used to visualizen-dimensional s-Pareto frontiers, and (iv) s-Pareto frontier-based approaches for considering uncertainty in concept selection. Simple structural examples are presented that illustrate representative applications of the proposed method.     

Generating Well-Distributed Sets of Pareto Points for Engineering Design Using Physical Programming

Engineering design generally involves two, possibly integrated, phases: (i) generating design options, and (ii) choosing the most satisfactory option on the basis of some determined criteria. The depth, or lack, of integration between these two phases defines different design approaches, and differing philosophical views from the part of researchers in the field of computational design. Optimization-Based Design (OBD) covers the spectrum of this depth of integration. While most OBD approaches strongly integrate these two phases, some employ computational optimization only in the first or second phase. Regardless of where a method or researcher lies in this philosophical spectrum, some requisite characteristics are fundamental to the effectiveness of OBD methods. In particular, (i) the Aggregate Objective Function (AOF) used in the optimization must have the ability to generate all Pareto solutions, (ii) the generation of any existing Pareto solutions must be possible with reasonable ease, and (iii) even changes in the AOF parameters should yield a well distributed set of Pareto solutions. This paper examines the effectiveness of physical programming (PP) with respect to the latter, yielding favorable conclusions. Previous papers have led to similarly positive conclusions with respect to the former two. This paper also presents a comparative study featuring PP and other popular methods, where PP is shown to perform favorably. A PP-based method for generating the Pareto frontier is presented.     

Local Pareto approximation for multi-objective optimization

The design process of complex systems often resorts to solving an optimization problem, which involves different disciplines and where all design criteria have to be optimized simultaneously. Mathematically, this problem can be reduced to a vector optimization problem. The solution of this problem is not unique and is represented by a Pareto surface in the objective function space. Once a Pareto solution is obtained, it may be very useful for the decision-maker to be able to perform a quick local approximation in the vicinity of this Pareto solution for sensitivity analysis. In this article, new linear and quadratic local approximations of the Pareto surface are derived and compared to existing formulas. The case of non-differentiable Pareto points (solutions) in the objective space is also analysed. The concept of a local quick Pareto analyser based on local sensitivity analysis is proposed. This Pareto analysis provides a quantitative insight into the relation between variations of the different objective functions under constraints. A few examples are considered to illustrate the concept and its advantages.     

ROBUSTNESS OPTIMIZATION FOR CONSTRAINED NONLINEAR PROGRAMMING PROBLEMS∗

In realistic situations engineering designs should take into consideration random aberrations from the stipulated design variables arising from manufacturing variability. Moreover, many environmental parameters are often stochastic in nature. Traditional nonlinear optimization attempts to find a deterministic optimum of a cost function and does not take into account the effect of these random variations on the objective. This paper attempts to devise a technique for finding optima of constrained nonlinear functions that are robust with respect to such variations. The expectation of the function over a domain of aberrations in the parameters is taken as a measure of ‘robustness’ of the function value at a point. It is pointed out that robustness optimization is ideally an attempt to trade off between ‘optimality’ and ‘robustness’. A newly-developed multi-criteria optimization technique known as Normal-Boundary Intersection is used to find evenly-spaced points on the Pareto curve for the ‘optimality’ and ‘robustness’ criteria. This Pareto curve enables the user to make the trade-off decision explicitly, free of arbitrary ‘weighting’ parameters.

This paper also formulates a derivative-based approximation for evaluating the expected value of the objective function on the nonlinear manifold defined by the state equations for the system. Existing procedures for evaluating the expectation usually involve numerical integration techniques requiring many solutions of the state equations for one evaluation of the expectation. The procedure presented here bypasses the need for multiple solutions of the state equations and hence provides a cheaper and more easily optimizable approximation to the expectation. Finally, this paper discusses how nonlinear inequality constraints should be treated in the presence of random parameters in the design. Computational results are presented for finding a robust optimum of a nonlinear structural optimization problem.     

Level Diagrams analysis of Pareto Front for multiobjective system redundancy allocation

Reliability-based and risk-informed design, operation, maintenance and regulation lead to multiobjective (multicriteria) optimization problems. In this context, the Pareto Front and Set found in a multiobjective optimality search provide a family of solutions among which the decision maker has to look for the best choice according to his or her preferences. Efficient visualization techniques for Pareto Front and Set analyses are needed for helping decision makers in the selection task.In this paper, we consider the multiobjective optimization of system redundancy allocation and use the recently introduced Level Diagrams technique for graphically representing the resulting Pareto Front and Set. Each objective and decision variable is represented on separate diagrams where the points of the Pareto Front and Set are positioned according to their proximity to ideally optimal points, as measured by a metric of normalized objective values. All diagrams are synchronized across all objectives and decision variables. On the basis of the analysis of the Level Diagrams, we introduce a procedure for reducing the number of solutions in the Pareto Front; from the reduced set of solutions, the decision maker can more easily identify his or her preferred solution.     

层合板频率响应与声功率双目标优化设计

利用遗传算法对复合材料层合板结构的固有频率间隔和辐射声功率进行双目标优化设计。利用分层理论结合有限元模型求解层合板的固有频率和振速分布。通过声辐射模态理论,计算层合板结构辐射声功率。以铺设角度作为设计变量,第一阶与第二阶固有频率间隔和辐射声功率作为双目标优化目标函数,以某4层的层合板结构为例,采用目标加权法优化目标函数。研究了不同权重系数、不同频率时固有频率间隔最大化和声功率最小化对应的优化铺设角度。由数值分析结果可知:不同的权重系数比下获得的Pareto最优解不同;在同一权重系数下,两个优化目标所起的作用不尽相同;随着频率的增加,选择相对较大的权重系数可使Pareto最优解较好地兼顾两个优化目标。     

Multi-objective analysis of a component-based representation within an interactive evolutionary design system

This article describes research relating to a user-centered evolutionary design system that evaluates both engineering and aesthetic aspects of design solutions during early-stage conceptual design. The experimental system comprises several components relating to user interaction, problem representation, evolutionary search and exploration and online learning. The main focus of the article is the evolutionary aspect of the system when using a single quantitative objective function plus subjective judgment of the user. Additionally, the manner in which the user-interaction aspect affects system output is assessed by comparing Pareto frontiers generated with and without user interaction via a multi-objective evolutionary algorithm (MOEA). A solution clustering component is also introduced and it is shown how this can improve the level of support to the designer when dealing with a complex design problem involving multiple objectives. Supporting results are from the application of the system to the design of urban furniture which, in this case, largely relates to seating design.     

A Swarm Metaphor for Multiobjective Design Optimization

This paper presents a new optimization algorithm to solve multiobjective design optimization problems based on behavioral concepts similar to that of a real swarm. The individuals of a swarm update their flying direction through communication with their neighboring leaders with an aim to collectively attain a common goal. The success of the swarm is attributed to three fundamental processes: identification of a set of leaders, selection of a leader for information acquisition, and finally a meaningful information transfer scheme. The proposed algorithm mimics the above behavioral processes of a real swarm. The algorithm employs a multilevel sieve to generate a set of leaders, a probabilistic crowding radius-based strategy for leader selection and a simple generational operator for information transfer. Two test problems, one with a discontinuous Pareto front and the other with a multi-modal Pareto front is solved to illustrate the capabilities of the algorithm in handling mathematically complex problems. Three well-studied engineering design optimization problems (unconstrained and constrained problems with continuous and discrete variables) are solved to illustrate the efficiency and applicability of the algorithm for multiobjective design optimization. The results clearly indicate that the swarm algorithm is capable of generating an extended Pareto front, consisting of well spread Pareto points with significantly fewer function evaluations when compared to the nondominated sorting genetic algorithm (NSGA).     

Gradient-based multiobjective optimization using a distance constraint technique and point replacement

This paper proposes techniques to improve the diversity of the searching points during the optimization process in an Aggregative Gradient-based Multiobjective Optimization (AGMO) method, so that well-distributed Pareto solutions are obtained. First to be discussed is a distance constraint technique, applied among searching points in the objective space when updating design variables, that maintains a minimum distance between the points. Next, a scheme is introduced that deals with updated points that violate the distance constraint, by deleting the offending points and introducing new points in areas of the objective space where searching points are sparsely distributed. Finally, the proposed method is applied to example problems to illustrate its effectiveness.     

The influence of proper efficiency on optimal solutions of scalarizing problems in multicriteria optimization

We investigate scalarizations for the determination of Pareto optima of multicriteria optimization problems which deliver properly efficient points in the sense of Geoffrion. Proper efficiency in the sense of Schönfeld is generalized by the simultaneous consideration of several weighted sums of the objective functions. This problem is geometrically interpreted by means of a polyhedral cone. The specialization of the parameters induces an extension of the weighted Chebyshev norm minimization for which we prove conditions for the existence of optimal solutions and statements confirming known more special results.     

飞机总体设计中的一种多目标鲁棒优化方法

 提出一种基于灵敏度的多目标鲁棒优化方法。针对各维设计变量存在扰动的情况,在原约束多目标优化模型上,附加偏差目标函数,并采用最差估计法对约束条件进行鲁棒可行性调整。采用全局敏度方程方法来计算目标函数和约束函数对设计变量的敏度,进而采用Pareto遗传算法搜索约束多目标优化问题的非劣解集,设计者可以根据不同的设计准则从中选择合适的设计点。将上述方法用于飞机总体参数优化设计,并与采用常规优化方法所得的优化结果进行了分析和比较。     

A new achievement scalarizing function based on parameterization in multiobjective optimization

This paper addresses a general multiobjective optimization problem. One of the most widely used methods of dealing with multiple conflicting objectives consists of constructing and optimizing a so-called achievement scalarizing function (ASF) which has an ability to produce any Pareto optimal or weakly/properly Pareto optimal solution. The ASF minimizes the distance from the reference point to the feasible region, if the reference point is unattainable, or maximizes the distance otherwise. The distance is defined by means of some specific kind of a metric introduced in the objective space. The reference point is usually specified by a decision maker and contains her/his aspirations about desirable objective values. The classical approach to constructing an ASF is based on using the Chebyshev metric L _∞. Another possibility is to use an additive ASF based on a modified linear metric L ₁. In this paper, we propose a parameterized version of an ASF. We introduce an integer parameter in order to control the degree of metric flexibility varying from L ₁ to L _∞. We prove that the parameterized ASF supports all the Pareto optimal solutions. Moreover, we specify conditions under which the Pareto optimality of each solution is guaranteed. An illustrative example for the case of three objectives and comparative analysis of parameterized ASFs with different values of the parameter are given. We show that the parameterized ASF provides the decision maker with flexible and advanced tools to detect Pareto optimal points, especially those whose detection with other ASFs is not straightforward since it may require changing essentially the reference point or weighting coefficients as well as some other extra computational efforts.     

Practical solutions for multi-objective optimization: An application to system reliability design problems

For multiple-objective optimization problems, a common solution methodology is to determine a Pareto optimal set. Unfortunately, these sets are often large and can become difficult to comprehend and consider. Two methods are presented as practical approaches to reduce the size of the Pareto optimal set for multiple-objective system reliability design problems. The first method is a pseudo-ranking scheme that helps the decision maker select solutions that reflect his/her objective function priorities. In the second approach, we used data mining clustering techniques to group the data by using the k-means algorithm to find clusters of similar solutions. This provides the decision maker with just k general solutions to choose from. With this second method, from the clustered Pareto optimal set, we attempted to find solutions which are likely to be more relevant to the decision maker. These are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. To demonstrate how these methods work, the well-known redundancy allocation problem was solved as a multiple objective problem by using the NSGA genetic algorithm to initially find the Pareto optimal solutions, and then, the two proposed methods are applied to prune the Pareto set.     

On the use of metamodel-assisted,multi-objective evolutionary algorithms

This article is concerned with the optimal use of metamodels in the context of multi-objective evolutionary algorithms which are based on computationally expensive function evaluations. The goal is to capture Pareto fronts of optimal solutions with the minimum computational cost. In each generation during the evolution, the metamodels act as filters that distinguish the most promising individuals, which will solely undergo exact and costly evaluations. By means of the so-called inexact pre-evaluation phase, based on continuously updated local metamodels, most of the non-promising individuals are put aside without aggravating the overall cost. The gain achieved through this technique is amazing in single-objective problems. However, with more than one objective, noticeable performance degradation occurs. This article scrutinizes the role of metamodels in multi-objective evolutionary algorithms and proposes ways to overcome expected weaknesses and improve their performance. Minimization of mathematical functions as well as aerodynamic shape optimization problems are used for demonstration purposes.     

A TABOO SEARCH BASED APPROACH TO FIND THE PARETO OPTIMAL SET IN MULTIPLE OBJECTIVE OPTIMIZATION

Taboo search is a heuristic optimization technique which works with a neighbourhood of solutions to optimize a given objective function. It is generally applied to single objective optimization problems. Taboo search has the potential for solving multiple objective optimization (MOO) problems, because it works with more than one solution at a time, and this gives it the opportunity to evaluate multiple objective functions simultaneously. In this paper, a taboo search based algorithm is developed to find Pareto optimal solutions in multiple objective optimization problems. The developed algorithm has been tested with a number of problems and compared with other techniques. Results obtained from this work have proved that a taboo search based algorithm can find Pareto optimal solutions in MOO effectively.