Showing papers on "Multi-objective optimization published in 2001"

Multi-Objective Optimization Using Evolutionary Algorithms

Abstract: From the Publisher: Evolutionary algorithms are relatively new, but very powerful techniques used to find solutions to many real-world search and optimization problems. Many of these problems have multiple objectives, which leads to the need to obtain a set of optimal solutions, known as effective solutions. It has been found that using evolutionary algorithms is a highly effective way of finding multiple effective solutions in a single simulation run. · Comprehensive coverage of this growing area of research · Carefully introduces each algorithm with examples and in-depth discussion · Includes many applications to real-world problems, including engineering design and scheduling · Includes discussion of advanced topics and future research · Features exercises and solutions, enabling use as a course text or for self-study · Accessible to those with limited knowledge of classical multi-objective optimization and evolutionary algorithms The integrated presentation of theory, algorithms and examples will benefit those working and researching in the areas of optimization, optimal design and evolutionary computing. This text provides an excellent introduction to the use of evolutionary algorithms in multi-objective optimization, allowing use as a graduate course text or for self-study.

PESA-II: region-based selection in evolutionary multiobjective optimization

Abstract: We describe a new selection technique for evolutionary multiobjective optimization algorithms in which the unit of selection is a hyperbox in objective space. In this technique, instead of assigning a selective fitness to an individual, selective fitness is assigned to the hyperboxes in objective space which are currently occupied by at least one individual in the current approximation to the Pareto frontier. A hyperbox is thereby selected, and the resulting selected individual is randomly chosen from this hyperbox. This method of selection is shown to be more sensitive to ensuring a good spread of development along the Pareto frontier than individual-based selection. The method is implemented in a modern multiobjective evolutionary algorithm, and performance is tested by using Deb's test suite of `T' functions with varying properties. The new selection technique is found to give significantly superior results to the other methods compared, namely PAES, PESA, and SPEA; each is a modern multi-objective optimization algorithm previously found to outperform earlier approaches on various problems.

Portfolio optimization with conditional value-at-risk objective and constraints

Abstract: Recently, a new approach for optimization of Conditional Value-at-Risk (CVaR) was suggested and tested with several applications. For continuous distributions, CVaR is defined as the expected loss exceeding Value-at Risk (VaR). However, generally, CVaR is the weighted average of VaR and losses exceeding VaR. Central to the approach is an optimization technique for calculating VaR and optimizing CVaR simultaneously. This paper extends this approach to the optimization problems with CVaR constraints. In particular, the approach can be used for maximizing expected returns under CVaR constraints. Multiple CVaR constraints with various confidence levels can be used to shape the profit/loss distribution. A case study for the portfolio of S&P 100 stocks is performed to demonstrate how the new optimization techniques can be implemented.

PDE: a Pareto-frontier differential evolution approach for multi-objective optimization problems

Abstract: The use of evolutionary algorithms (EAs) to solve problems with multiple objectives (known as multi-objective optimization problems (MOPs)) has attracted much attention. Being population based approaches, EAs offer a means to find a group of Pareto-optimal solutions in a single run. Differential evolution (DE) is an EA that was developed to handle optimization problems over continuous domains. The objective of this paper is to introduce a novel Pareto-frontier differential evolution (PDE) algorithm to solve MOPs. The solutions provided by the proposed algorithm for two standard test problems, outperform the Strength Pareto Evolutionary Algorithm, one of the state-of-the-art evolutionary algorithms for solving MOPs.

A Micro-Genetic Algorithm for Multiobjective Optimization

Abstract: In this paper, we propose a multiobjective optimization approach based on a micro genetic algorithm (micro-GA) which is a genetic algorithm with a very small population (four individuals were used in our experiment) and a reinitialization process. We use three forms of elitism and a memory to generate the initial population of the micro-GA. Our approach is tested with several standard functions found in the specialized literature. The results obtained are very encouraging, since they show that this simple approach can produce an important portion of the Pareto front at a very low computational cost.

Reducing Local Optima in Single-Objective Problems by Multi-objectivization

Abstract: One common characterization of how simple hill-climbing optimization methods can fail is that they become trapped in local optima - a state where no small modification of the current best solution will produce a solution that is better. This measure of 'better' depends on the performance of the solution with respect to the single objective being optimized. In contrast, multi-objective optimization (MOO) involves the simultaneous optimization of a number of objectives. Accordingly, the multi-objective notion of 'better' permits consideration of solutions that may be superior in one objective but not in another. Intuitively, we may say that this gives a hill-climber in multi-objective space more freedom to explore and less likelihood of becoming trapped. In this paper, we investigate this intuition by comparing the performance of simple hill-climber-style algorithms on single-objective problems and multi-objective versions of those same problems. Using an abstract building-block problem we illustrate how 'multi-objectivizing' a single-objective optimization (SOO) problem can remove local optima. Then we investigate small instances of the travelling salesman problem where additional objectives are defined using arbitrary sub-tours. Results indicate that multi-objectivization can reduce local optima and facilitate improved optimization in some cases. These results enlighten our intuitions about the nature of search in multi-objective optimization and sources of difficulty in single-objective optimization.

Scalable test problems for evolutionary multi-objective optimization

Abstract: After adequately demonstrating the ability to solve different two-objective optimization problems, multi-objective evolutionary algorithms (MOEAs) must now show their efficacy in handling problems having more than two objectives. In this paper, we have suggested three different approaches for systematically designing test problems for this purpose. The simplicity of construction, scalability to any number of decision variables and objectives, knowledge of exact shape and location of the resulting Pareto-optimal front, and introduction of controlled difficulties in both converging to the true Pareto-optimal front and maintaining a widely distributed set of solutions are the main features of the suggested test problems. Because of the above features, they should be found useful in various research activities on MOEAs, such as testing the performance of a new MOEA, comparing different MOEAs, and better understanding of the working principles of MOEAs.

Evolutionary Algorithms for Single and Multicriteria Design Optimization

Abstract: Introduction to Design Optimization.- Genetic and Evolutionary Algorithms as a Design Optimization Tool.- Advanced Evolutionary Algorithm Techniques.- Evolutionary Algorithms for Single Criterion Optimization.- Evolutionary Algorithms for Multicriteria Optimization.- Some Other Evolutionary Algorithms Based Methods.- Design Optimization Examples and Their Solution by Evolutionary Algorithms.- Appendix: Evolutionary Optimization System.- Appendix: C Codes for Two Design Optimization.

Evolutionary algorithms with dynamic population size and local exploration for multiobjective optimization

Abstract: Evolutionary algorithms have been recognized to be well suited for multiobjective optimization. These methods, however, need to "guess" for an optimal constant population size in order to discover the usually sophisticated tradeoff surface. This paper addresses the issue by presenting a novel incrementing multiobjective evolutionary algorithm (IMOEA) with dynamic population size that is computed adaptively according to the online discovered tradeoff surface and its desired population distribution density. It incorporates the method of fuzzy boundary local perturbation with interactive local fine tuning for broader neighborhood exploration. This achieves better convergence as well as discovering any gaps or missing tradeoff regions at each generation. Other advanced features include a proposed preserved strategy to ensure better stability and diversity of the Pareto front and a convergence representation based on the concept of online population domination to provide useful information. Extensive simulations are performed on two benchmark and one practical engineering design problems.

Bi-Criterion Optimization with Multi Colony Ant Algorithms

Abstract: In this paper we propose a new approach to solve bi-criterion optimization problems with ant algorithms where several colonies of ants cooperate in finding good solutions. We introduce two methods for co-operation between the colonies and compare them with a multistart ant algorithm that corresponds to the case of no cooperation. Heterogeneous colonies are used in the algorithm, i.e. the ants differ in their preferences between the two criteria. Every colony uses two pheromone matrices -- each suitable for one optimization criterion. As a test problem we use the Single Machine Total Tardiness problem with changeover costs.

Pareto-Front Exploration with Uncertain Objectives

Abstract: We consider the problem of exploration of the set of all global optima (Pareto-points) or an approximation thereof in the context of multi-objective function optimization. Up to now, set oriented techniques assume that the evaluation of the m-dimensional vector of objectives can be done exactly which is important to steer the search process towards global optima. Here, we extend such techniques to allow objectives to be uncertain, i.e., vary within intervals. This may be often the case if the exact computation of objectives is computationally too expensive such that only estimates on the objective values of a design point may be derived. For objective values that are constrained by intervals, we derive a theory of probabilistic dominance, an extension of the definition of Pareto-dominance. Also, we show how this theory may be used in order to guide the selection process to approximate the Pareto-set.

A survey of multiobjective optimization in engineering design

Abstract: Real world engineering design problems are usually characterized by the presence of many conflicting objectives. Therefore, it is natural to look at the engineering design problem as a multiobjective optimization problem. This report summarizes a survey of techniques to conduct multiobjective optimization in an engineering design context. The report starts with discussing some of the difficulties of expressing the value of a deign and how to characterize different design variables. Thereafter we look more closely on the design problem in order to reformulate the design problem as a multiobjective optimization problem. As engineering design problems often consist of a mixture of numerical simulations, analytical calculations and catalog selections, there is no easy way of calculating derivatives of the objectives function. Therefore, non-gradient optimization methods are better suited for these types of problems. Different types of non-gradient method are discussed in the report and different ways of developing hybrid methods are presented as well. As most optimization problems are multiobjective to there nature, there are many methods available to tackle these kind of problems. Generally, a multiobjective optimization problem can be handled in four different ways depending on when the decision-maker articulates his or her preference on the different objectives; never, before, during or after the actual optimization procedure. The most common way is to aggregate the different objectives to one figure of merit by using a weighted sum and the conduct the actual optimization. There is however an abundance of other ways in which multiobjective optimization can be conducted, some of them are presented in this report. A survey of multiobjective optimization methods in engineering design

Multiobjective design optimization by an evolutionary algorithm

Abstract: This paper presents an evolutionary algorithm for generic multiobjective design optimization problems. The algorithm is based on nondominance of solutions in the objective and constraint space and uses effective mating strategies to improve solutions that are weak in either. Since the methodology is based on nondominance, scaling and aggregation affecting conventional penalty function methods for constraint handling does not arise. The algorithm incorporates intelligent partner selection for cooperative mating. The diversification strategy is based on niching which results in a wide spread of solutions in the parametric space. Results of the algorithm for the design examples clearly illustrate the efficiency of the algorithm in solving multidisciplinary design optimization problems.

A Short Tutorial on Evolutionary Multiobjective Optimization

Abstract: This tutorial will review some of the basic concepts related to evolutionary multiobjective optimization (i.e., the use of evolutionary algorithms to handle more than one objective function at a time). The most commonly used evolutionary multiobjective optimization techniques will be described and criticized, including some of their applications. Theory, test functions and metrics will be also discussed. Finally, we will provide some possible paths of future research in this area.

Dynamic Weighted Aggregation for evolutionary multi-objective optimization: why does it work and how?

Abstract: Evolutionary Dynamic Weighted Aggregation (EDWA) has shown to be both effective and computationally efficient [1] for multi-objective optimization (MOO). Besides, it was also found empirically and surprisingly that EDWA was able to deal with multi-objective optimization problems with a concave Pareto front, which has proved to be beyond the capability of the Conventional Weighted Aggregation (CWA) methods [2]. In this paper, a theory on why CWA fails for multi-objective problems with a concave Pareto front is provided schematically. According to this theory, it can easily be explained why EDWA has worked well for both convex and concave multi-objective problems. Simulation examples are conducted on various test functions to support our theory. It is concluded that EDWA is an effective and efficient method for solving multi-objective optimization problems.

Multiobjective optimization using a micro-genetic algorithm

Abstract: In this paper, we propose a micro genetic algorithm with three forms of elitism for multiobjective optimization. We show how this relatively simple algorithm coupled with an external file and a diversity approach based on geographical distribution can generate efficiently the Pareto fronts of several difficult test functions (both constrained and unconstrained). A metric based on the average distance to the Pareto optimal set is used to compare our results against two evolutionary multiobjective optimization techniques recently proposed in the literature.

Efficient Pareto Frontier Exploration using Surrogate Approximations

Abstract: 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.

A Hybrid Multi-objective Evolutionary Approach to Engineering Shape Design

Abstract: Evolutionary optimization algorithms work with a population of solutions, instead of a single solution. Since multi-objective optimization problems give rise to a set of Pareto-optimal solutions, evolutionary optimization algorithms are ideal for handling multi-objective optimization problems. Over many years of research and application studies have produced a number of efficient multi-objective evolutionary algorithms (MOEAs), which are ready to be applied to real-world problems. In this paper, we propose a practical approach, which will enable an user to move closer to the true Pareto-optimal front and simultaneously reduce the size of the obtained non-dominated solution set. The efficacy of the proposed approach is demonstrated in solving a number of mechanical shape optimization problems, including a simply-supported plate design, a cantilever plate design, a hoister design, and a bicycle frame design. The results are interesting and suggest immediate application of the proposed technique in more complex engineering design problems.

Nonlinear goal programming using multi-objective genetic algorithms

Abstract: Goal programming is a technique often used in engineering design activities primarily to find a compromised solution which will simultaneously satisfy a number of design goals. In solving goal programming problems, classical methods reduce the multiple goal-attainment problem into a single objective of minimizing a weighted sum of deviations from goals. This procedure has a number of known difficulties. First, the obtained solution to the goal programming problem is sensitive to the chosen weight vector. Second, the conversion to a single-objective optimization problem involves additional constraints. Third, since most real-world goal programming problems involve nonlinear criterion functions, the resulting single-objective optimization problem becomes a nonlinear programming problem, which is difficult to solve using classical optimization methods. In tackling nonlinear goal programming problems, although successive linearization techniques have been suggested, they are found to be sensitive to the chosen starting solution. In this paper, we pose the goal programming problem as a multi-objective optimization problem of minimizing deviations from individual goals and then suggest an evolutionary optimization algorithm to find multiple Pareto-optimal solutions of the resulting multi-objective optimization problem. The proposed approach alleviates all the above difficulties. It does not need any weight vector. It eliminates the need of having extra constraints needed with the classical formulations. The proposed approach is also suitable for solving goal programming problems having nonlinear criterion functions and having a non-convex trade-off region. The efficacy of the proposed approach is demonstrated by solving a number of nonlinear goal programming test problems and an engineering design problem. In all problems, multiple solutions (each corresponding to a different weight vector) to the goal programming problem are found in one single simulation run. The results suggest that the proposed approach is an effective and practical tool for solving real-world goal programming problems.

Adapting Weighted Aggregation for Multiobjective Evolution Strategies

Abstract: The conventional weighted aggregation method is extended to realize multi-objective optimization. The basic idea is that systematically changing the weights during evolution will lead the population to the Pareto front. Two possible methods are investigated. One method is to assign a uniformly distributed random weight to each individual in the population in each generation. The other method is to change the weight periodically with the process of the evolution. We found in both cases that the population is able to approach the Pareto front, although it will not keep all the found Pareto solutions in the population. Therefore, an archive of non-dominated solutions is maintained. Case studies are carried out on some of the test functions used in [1] and [2]. Simulation results show that the proposed approaches are simple and effective.

Pareto Optimality in Coevolutionary Learning

Abstract: We develop a novel coevolutionary algorithm based upon the concept of Pareto optimality. The Pareto criterion is core to conventional multi-objective optimization (MOO) algorithms. We can think of agents in a coevolutionary system as performing MOO, as well: An agent interacts with many other agents, each of which can be regarded as an objective for optimization. We adapt the Pareto concept to allow agents to follow gradient and create gradient for others to follow, such that co-evolutionary learning succeeds. We demonstrate our Pareto coevolution methodology with the majority function, a density classification task for cellular automata.

Some Methods for Nonlinear Multi-objective Optimization

Abstract: A general overview of nonlinear multiobjective optimization methods is given. The basic features of several methods are introduced so that an appropriate method could be found for different purposes. The methods are classified according to the role of a decision maker in the solution process. The main emphasis is devoted to interactive methods where the decision maker progressively provides preference information so that the most satisfactory solution can be found.

Multiobjective query optimization

Abstract: The optimization of queries in distributed database systems is known to be subject to delicate trade-offs. For example, the Mariposa database system allows users to specify a desired delay-cost tradeoff (that is, to supply a decreasing function u(d), specifying how much the user is willing to pay in order to receive the query results within time d); Mariposa divides a query graph into horizontal “strides,” analyzes each stride, and uses a greedy heuristic to find the “best” plan for all strides. We show that Mariposa's greedy heuristic can be arbitrarily far from the desired optimum. Applying a recent approach in multiobjective optimization algorithms to this problem, we show that the optimum cost-delay trade-off (Pareto) curve in Mariposa's framework can be approximated fast within any desired accuracy. We also present a polynomial algorithm for the general multiobjective query optimization problem, which approximates arbirarily well the optimum cost-delay tradeoff (without the restriction of Mariposa's heuristic stride subdivision).

Reliability and Costs Optimization for Distribution Networks Expansion Using an Evolutionary Algorithm

Abstract: This paper presents a multiobjective optimization methodology, using an evolutionary algorithm, for finding out the best distribution network reliability while simultaneously minimizing the system expansion costs. A nonlinear mixed integer optimization model, achieving the optimal sizing and location of future feeders (reserve feeders and operation feeders) and substations, has been used. The proposed methodology has been tested intensively for distribution systems with dimensions that are significantly larger than the ones frequently found in the papers about this issue. Furthermore, this methodology is general since it is suitable for the multiobjective optimization of n objectives simultaneously. The algorithm can determine the set of optimal nondominated solutions, allowing the planner to obtain the optimal locations and sizes of the reserve feeders that achieve the best system reliability with the lowest expansion costs. The model and the algorithm have been applied intensively to real life power systems showing its potential of applicability to large distribution networks in practice.

Pareto analysis in multiobjective optimization using the collinearity theorem and scaling method

Abstract: This paper presents a method to predict the relative objective weighting scheme necessary to cause arbitrary members of a Pareto solution set to become optimal. First, a polynomial description of the Pareto set is constructed utilizing simulation and high performance computing. Then, using geometric relationships between the member of the Pareto set in question, the location of the utopia point and the polynomial coefficients, the weighting of the performance metrics which causes a particular member of the Pareto set to become optimal is determined. The use of this technique, termed the scaling method, is examined via using a sample problem from the field of vehicle dynamics optimization. The scaling method is based on the collinearity theorem which is also presented in the paper.