Showing papers on "Multi-objective optimization published in 2006"

ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems

Abstract: This paper concerns multiobjective optimization in scenarios where each solution evaluation is financially and/or temporally expensive. We make use of nine relatively low-dimensional, nonpathological, real-valued functions, such as arise in many applications, and assess the performance of two algorithms after just 100 and 250 (or 260) function evaluations. The results show that NSGA-II, a popular multiobjective evolutionary algorithm, performs well compared with random search, even within the restricted number of evaluations used. A significantly better performance (particularly, in the worst case) is, however, achieved on our test set by an algorithm proposed herein-ParEGO-which is an extension of the single-objective efficient global optimization (EGO) algorithm of Jones et al. ParEGO uses a design-of-experiments inspired initialization procedure and learns a Gaussian processes model of the search landscape, which is updated after every function evaluation. Overall, ParEGO exhibits a promising performance for multiobjective optimization problems where evaluations are expensive or otherwise restricted in number.

The MOORA method and its application to privatization in a transition economy

Abstract: Abstract: A new method is proposed for multi-objective optimization with discrete alternatives: MOORA (Multi-Objective Optimization on the basis of Ratio Analysis). This method refers to a matrix of responses of alternatives to objectives, to which ratios are applied. A well established other method for multi-objective optimization is used for comparison, namely the reference point method. Later on, it is demonstrated that this is the best choice among the different competing methods. In MOORA the set of ratios has the square roots of the sum of squared responses as denominators. These ratios, as dimensionless, seem to be the best choice among different ratios. These dimensionless ratios, situated between zero and one, are added in the case of maximization or subtracted in case of minimization. Finally, all alternatives are ranked, according to the obtained ratios. Eventually, to give more importance to an objective, an objective can be replaced by different sub-objectives or a coefficient of importance can be specified. An example on privatization in a transition economy illustrates the application of the method. If application is situated originally in a “welfare” economy, centered on production, MOORA becomes even more significant in a “wellbeing economy”, where consumer sovereignty is assumed.

Reference point based multi-objective optimization using evolutionary algorithms

Abstract: Evolutionary multi-objective optimization (EMO) methodologies have been amply applied to find a representative set of Pareto-optimal solutions in the past decade and beyond. Although there are advantages of knowing the range of each objective for Pareto-optimality and the shape of the Pareto-optimal frontier itself in a problem for an adequate decision-making, the task of choosing a single preferred Pareto-optimal solution is also an important task which has received a lukewarm attention so far. In this paper, we combine one such preference based strategy with an EMO methodology and demonstrate how, instead of one solution, a preferred set solutions near the reference points can be found parallely. We propose a modified EMO procedure based on the elitist non-dominated sorting GAor NSGA-II. On two-objective to 10-objective optimization problems, the modified NSGA-II approach shows its efficacy in finding an adequate set of Pareto-optimal points. Such procedures will provide the decision-maker with a set of solutions near her/his preference so that a better and a more reliable decision can be made.

Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels

Abstract: This paper presents and analyzes in detail an efficient search method based on evolutionary algorithms (EA) assisted by local Gaussian random field metamodels (GRFM). It is created for the use in optimization problems with one (or many) computationally expensive evaluation function(s). The role of GRFM is to predict objective function values for new candidate solutions by exploiting information recorded during previous evaluations. Moreover, GRFM are able to provide estimates of the confidence of their predictions. Predictions and their confidence intervals predicted by GRFM are used by the metamodel assisted EA. It selects the promising members in each generation and carries out exact, costly evaluations only for them. The extensive use of the uncertainty information of predictions for screening the candidate solutions makes it possible to significantly reduce the computational cost of singleand multiobjective EA. This is adequately demonstrated in this paper by means of mathematical test cases and a multipoint airfoil design in aerodynamics

Multiobjective evolutionary algorithms for electric power dispatch problem

Abstract: The potential and effectiveness of the newly developed Pareto-based multiobjective evolutionary algorithms (MOEA) for solving a real-world power system multiobjective nonlinear optimization problem are comprehensively discussed and evaluated in this paper. Specifically, nondominated sorting genetic algorithm, niched Pareto genetic algorithm, and strength Pareto evolutionary algorithm (SPEA) have been developed and successfully applied to an environmental/economic electric power dispatch problem. A new procedure for quality measure is proposed in this paper in order to evaluate different techniques. A feasibility check procedure has been developed and superimposed on MOEA to restrict the search to the feasible region of the problem space. A hierarchical clustering algorithm is also imposed to provide the power system operator with a representative and manageable Pareto-optimal set. Moreover, an approach based on fuzzy set theory is developed to extract one of the Pareto-optimal solutions as the best compromise one. These multiobjective evolutionary algorithms have been individually examined and applied to the standard IEEE 30-bus six-generator test system. Several optimization runs have been carried out on different cases of problem complexity. The results of MOEA have been compared to those reported in the literature. The results confirm the potential and effectiveness of MOEA compared to the traditional multiobjective optimization techniques. In addition, the results demonstrate the superiority of the SPEA as a promising multiobjective evolutionary algorithm to solve different power system multiobjective optimization problems.

A survey of simulated annealing as a tool for single and multiobjective optimization

Abstract: This paper presents a comprehensive review of simulated annealing (SA)-based optimization algorithms. SA-based algorithms solve single and multiobjective optimization problems, where a desired global minimum/maximum is hidden among many local minima/maxima. Three single objective optimization algorithms (SA, SA with tabu search and CSA) and five multiobjective optimization algorithms (SMOSA, UMOSA, PSA, WDMOSA and PDMOSA) based on SA have been presented. The algorithms are briefly discussed and are compared. The key step of SA is probability calculation, which involves building the annealing schedule. Annealing schedule is discussed briefly. Computational results and suggestions to improve the performance of SA-based multiobjective algorithms are presented. Finally, future research in the area of SA is suggested.

Introducing robustness in multi-objective optimization

Abstract: In optimization studies including multi-objective optimization, the main focus is placed on finding the global optimum or global Pareto-optimal solutions, representing the best possible objective values. However, in practice, users may not always be interested in finding the so-called global best solutions, particularly when these solutions are quite sensitive to the variable perturbations which cannot be avoided in practice. In such cases, practitioners are interested in finding the robust solutions which are less sensitive to small perturbations in variables. Although robust optimization is dealt with in detail in single-objective evolutionary optimization studies, in this paper, we present two different robust multi-objective optimization procedures, where the emphasis is to find a robust frontier, instead of the global Pareto-optimal frontier in a problem. The first procedure is a straightforward extension of a technique used for single-objective optimization and the second procedure is a more practical approach enabling a user to set the extent of robustness desired in a problem. To demonstrate the differences between global and robust multi-objective optimization principles and the differences between the two robust optimization procedures suggested here, we develop a number of constrained and unconstrained test problems having two and three objectives and show simulation results using an evolutionary multi-objective optimization (EM0) algorithm. Finally, we also apply both robust optimization methodologies to an engineering design problem.

Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation

Abstract: This paper presents an adaptive weighted sum (AWS) method for multiobjective optimization problems. The method extends the previously developed biobjective AWS method to problems with more than two objective functions. In the first phase, the usual weighted sum method is performed to approximate the Pareto surface quickly, and a mesh of Pareto front patches is identified. Each Pareto front patch is then refined by imposing additional equality constraints that connect the pseudonadir point and the expected Pareto optimal solutions on a piecewise planar hypersurface in the \( {m} \)-dimensional objective space. It is demonstrated that the method produces a well-distributed Pareto front mesh for effective visualization, and that it finds solutions in nonconvex regions. Two numerical examples and a simple structural optimization problem are solved as case studies.

A Multiobjective Optimization-Based Evolutionary Algorithm for Constrained Optimization

Abstract: A considerable number of constrained optimization evolutionary algorithms (COEAs) have been proposed due to increasing interest in solving constrained optimization problems (COPs) by evolutionary algorithms (EAs). In this paper, we first review existing COEAs. Then, a novel EA for constrained optimization is presented. In the process of population evolution, our algorithm is based on multiobjective optimization techniques, i.e., an individual in the parent population may be replaced if it is dominated by a nondominated individual in the offspring population. In addition, three models of a population-based algorithm-generator and an infeasible solution archiving and replacement mechanism are introduced. Furthermore, the simplex crossover is used as a recombination operator to enrich the exploration and exploitation abilities of the approach proposed. The new approach is tested on 13 well-known benchmark functions, and the empirical evidence suggests that it is robust, efficient, and generic when handling linear/nonlinear equality/inequality constraints. Compared with some other state-of-the-art algorithms, our algorithm remarkably outperforms them in terms of the best, mean, and worst objective function values and the standard deviations. It is noteworthy that our algorithm does not require the transformation of equality constraints into inequality constraints

Innovization: innovating design principles through optimization

Abstract: This paper introduces a new design methodology (we call it "innovization") in the context of finding new and innovative design principles by means of optimization techniques. Although optimization algorithms are routinely used to find an optimal solution corresponding to an optimization problem, the task of innovization stretches the scope beyond an optimization task and attempts to unveil new, innovative, and important design principles relating to decision variables and objectives, so that a deeper understanding of the problem can be obtained. The variety of problems chosen in the paper and the resulting innovations obtained for each problem amply demonstrate the usefulness of the innovization task. The results should encourage a wide spread applicability of the proposed innovization procedure (which is not simply an optimization procedure) to other problem-solving tasks.

Dynamic multi-objective optimization with evolutionary algorithms: a forward-looking approach

Abstract: This work describes a forward-looking approach for the solution of dynamic (time-changing) problems using evolutionary algorithms. The main idea of the proposed method is to combine a forecasting technique with an evolutionary algorithm. The location, in variable space, of the optimal solution (or of the Pareto optimal set in multi-objective problems) is estimated using a forecasting method. Then, using this forecast, an anticipatory group of individuals is placed on and near the estimated location of the next optimum. This prediction set is used to seed the population when a change in the objective landscape arrives, aiming at a faster convergence to the new global optimum. The forecasting model is created using the sequence of prior optimum locations, from which an estimate for the next location is extrapolated. Conceptually this approach encompasses advantages of memory methods by making use of information available from previous time steps. Combined with a convergence/diversity balance mechanism it creates a robust algorithm for dynamic optimization. This strategy can be applied to single objective and multi-objective problems, however in this work it is tested on multi-objective problems. Initial results indicate that the approach improves algorithm performance, especially in problems where the frequency of objective change is high.

Combining Model-based and Genetics-based Offspring Generation for Multi-objective Optimization Using a Convergence Criterion

Abstract: In our previous work conducted by Aimin Zhou et. al., (2005), it has been shown that the performance of multi-objective evolutionary algorithms can be greatly enhanced if the regularity in the distribution of Pareto-optimal solutions is used. This paper suggests a new hybrid multi-objective evolutionary algorithm by introducing a convergence based criterion to determine when the model-based method and when the genetics-based method should be used to generate offspring in each generation. The basic idea is that the genetics-based method, i.e., crossover and mutation, should be used when the population is far away from the Pareto front and no obvious regularity in population distribution can be observed. When the population moves towards the Pareto front, the distribution of the individuals will show increasing regularity and in this case, the model-based method should be used to generate offspring. The proposed hybrid method is verified on widely used test problems and our simulation results show that the method is effective in achieving Pareto-optimal solutions compared to two state-of-the-art evolutionary multi-objective algorithms: NSGA-II and SPEA2, and our pervious method in Aimin Zhou et. al., (2005).

Modified Differential Evolution for Constrained Optimization

Abstract: In this paper, we present a Differential-Evolution based approach to solve constrained optimization problems. The aim of the approach is to increase the probability of each parent to generate a better offspring. This is done by allowing each solution to generate more than one offspring but using a different mutation operator which combines information of the best solution in the population and also information of the current parent to find new search directions. Three selection criteria based on feasibility are used to deal with the constraints of the problem and also a diversity mechanism is added to maintain infeasible solutions located in promising areas of the search space. The approach is tested in a set of test problems proposed for the special session on Constrained Real Parameter Optimization. The results obtained are discussed and some conclusions are established.

Optimal Reservoir Operation Using Multi-Objective Evolutionary Algorithm

Abstract: This paper presents a Multi-objective Evolutionary Algorithm (MOEA) to derive a set of optimal operation policies for a multipurpose reservoir system. One of the main goals in multi-objective optimization is to find a set of well distributed optimal solutions along the Pareto front. Classical optimization methods often fail in attaining a good Pareto front. To overcome the drawbacks faced by the classical methods for Multi-objective Optimization Problems (MOOP), this study employs a population based search evolutionary algorithm namely Multi-objective Genetic Algorithm (MOGA) to generate a Pareto optimal set. The MOGA approach is applied to a realistic reservoir system, namely Bhadra Reservoir system, in India. The reservoir serves multiple purposes irrigation, hydropower generation and downstream water quality requirements. The results obtained using the proposed evolutionary algorithm is able to offer many alternative policies for the reservoir operator, giving flexibility to choose the best out of them. This study demonstrates the usefulness of MOGA for a real life multi-objective optimization problem.

Are all objectives necessary? on dimensionality reduction in evolutionary multiobjective optimization

Abstract: Most of the available multiobjective evolutionary algorithms (MOEA) for approximating the Pareto set have been designed for and tested on low dimensional problems (≤3 objectives). However, it is known that problems with a high number of objectives cause additional difficulties in terms of the quality of the Pareto set approximation and running time. Furthermore, the decision making process becomes the harder the more objectives are involved. In this context, the question arises whether all objectives are necessary to preserve the problem characteristics. One may also ask under which conditions such an objective reduction is feasible, and how a minimum set of objectives can be computed. In this paper, we propose a general mathematical framework, suited to answer these three questions, and corresponding algorithms, exact and heuristic ones. The heuristic variants are geared towards direct integration into the evolutionary search process. Moreover, extensive experiments for four well-known test problems show that substantial dimensionality reductions are possible on the basis of the proposed methodology.

A distributed Cooperative coevolutionary algorithm for multiobjective optimization

Abstract: Recent advances in evolutionary algorithms show that coevolutionary architectures are effective ways to broaden the use of traditional evolutionary algorithms. This paper presents a cooperative coevolutionary algorithm (CCEA) for multiobjective optimization, which applies the divide-and-conquer approach to decompose decision vectors into smaller components and evolves multiple solutions in the form of cooperative subpopulations. Incorporated with various features like archiving, dynamic sharing, and extending operator, the CCEA is capable of maintaining archive diversity in the evolution and distributing the solutions uniformly along the Pareto front. Exploiting the inherent parallelism of cooperative coevolution, the CCEA can be formulated into a distributed cooperative coevolutionary algorithm (DCCEA) suitable for concurrent processing that allows inter-communication of subpopulations residing in networked computers, and hence expedites the computational speed by sharing the workload among multiple computers. Simulation results show that the CCEA is competitive in finding the tradeoff solutions, and the DCCEA can effectively reduce the simulation runtime without sacrificing the performance of CCEA as the number of peers is increased

Normalization and Other Topics in Multi­Objective Optimization

Abstract: A multi­objective optimization typically arises in various engineering modelling problems, financial applications, and other problems where the decision maker chooses among several competing objectives to satisfy (see, e.g. [5]). In this report we consider a multiobjective optimization problem that comes from the financial sector. However, the analysis is applicable to any problem that retains similar characteristics. In particular, we focus on techniques for the normalization of objective functions. The normalization plays an important role in ensuring the consistency of optimal solutions with the preferences expressed by the decision maker. We also compare several approaches to solve the problem assuming that a linear or a mixed integer programming solver, such as CPLEX, is available. Namely, we consider weighted sum and hierarchical or ε-constraint methods, see also [1, 3, 6]. Although this report tends to provide a general discussion about multi-objective optimization, the proposed analysis and the resulting algorithm are specifically aimed at practical implementation. The reader should keep in mind that different approaches may be more effective should one not be constrained by various factors that are induced by the environment, and that are not discussed within the scope of this paper. A multi-objective optimization problem can be written in the following form

Dynamic origin-destination demand estimation using automatic vehicle identification data

Abstract: This paper proposes a dynamic origin-destination (OD) estimation method to extract valuable point-to-point split-fraction information from automatic vehicle identification (AVI) counts without estimating market-penetration rates and identification rates of AVI tags. A nonlinear ordinary least-squares estimation model is presented to combine AVI counts, link counts, and historical demand information into a multiobjective optimization framework. A joint estimation formulation and a one-sided linear-penalty formulation are further developed to take into account possible identification and representativeness errors, and the resulting optimization problems are solved by using an iterative bilevel estimation procedure. Based on a synthetic data set, this study shows the effectiveness of the proposed estimation models under different market-penetration rates and identification rates

Multiobjective optimization and evolutionary algorithms for the application mapping problem in multiprocessor system-on-chip design

Abstract: Sesame is a software framework that aims at developing a modeling and simulation environment for the efficient design space exploration of heterogeneous embedded systems. Since Sesame recognizes separate application and architecture models within a single system simulation, it needs an explicit mapping step to relate these models for cosimulation. The design tradeoffs during the mapping stage, namely, the processing time, power consumption, and architecture cost, are captured by a multiobjective nonlinear mixed integer program. This paper aims at investigating the performance of multiobjective evolutionary algorithms (MOEAs) on solving large instances of the mapping problem. With two comparative case studies, it is shown that MOEAs provide the designer with a highly accurate set of solutions in a reasonable amount of time. Additionally, analyses for different crossover types, mutation usage, and repair strategies for the purpose of constraints handling are carried out. Finally, a number of multiobjective optimization results are simulated for verification.

Electric distribution network multiobjective design using a problem-specific genetic algorithm

Abstract: This paper presents a multiobjective approach for the design of electrical distribution networks. The objectives are defined as a monetary cost index (including installation cost and energy losses cost) and a system failure index. The true Pareto-optimal solutions are found with a multiobjective genetic algorithm that employs an efficient variable encoding scheme and some problem-specific mutation and crossover operators. Results based on 21- and 100-bus systems are presented. The information gained from the Pareto-optimal solution set is shown to be useful for the decision-making stage of distribution network evolution planning.

A discussion of scalarization techniques for multiple objective integer programming

Abstract: In this paper we consider solution methods for multiobjective integer programming (MOIP) problems based on scalarization. We define the MOIP, discuss some common scalarizations, and provide a general formulation that encompasses most scalarizations that have been applied in the MOIP context as special cases. We show that these methods suffer some drawbacks by either only being able to find supported efficient solutions or introducing constraints that can make the computational effort to solve the scalarization prohibitive. We show that Lagrangian duality applied to the general scalarization does not remedy the situation. We also introduce a new scalarization technique, the method of elastic constraints, which is shown to be able to find all efficient solutions and overcome the computational burden of the scalarizations that use constraints on objective values. Finally, we present some results from an application in airline crew scheduling as evidence.

jMetal: a Java Framework for Developing Multi-Objective Optimization Metaheuristics

Abstract: This paper introduces jMetal, an object-oriented Java-based framework aimed at facilitating the development of metaheuristics for solving multi-objective optimization problems (MOPs). jMetal provides a rich set of classes which can be used as the building blocks of multi-objective metaheuristics; thus, taking advantage of code-reusing, the algorithms share the same base components, such as implementations of genetic operators and density estimators, so making the fair comparison of different metaheuristics for MOPs possible. The framework also incorporates a significant set of problems used as a benchmark in many comparative studies. We have implemented a number of multi-objective metaheuristics using jMetal, and an evaluation of these techniques using our framework is presented.