Showing papers on "Multi-objective optimization published in 2004"
Book•
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TL;DR: In this article, the focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them, and a comprehensive introduction to the subject is given. But the focus of this book is not on the optimization problem itself, but on the problem of finding the appropriate technique to solve it.
Abstract: Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.
33,341 citations
TL;DR: An approach in which Pareto dominance is incorporated into particle swarm optimization (PSO) in order to allow this heuristic to handle problems with several objective functions and indicates that the approach is highly competitive and that can be considered a viable alternative to solve multiobjective optimization problems.
Abstract: This paper presents an approach in which Pareto dominance is incorporated into particle swarm optimization (PSO) in order to allow this heuristic to handle problems with several objective functions. Unlike other current proposals to extend PSO to solve multiobjective optimization problems, our algorithm uses a secondary (i.e., external) repository of particles that is later used by other particles to guide their own flight. We also incorporate a special mutation operator that enriches the exploratory capabilities of our algorithm. The proposed approach is validated using several test functions and metrics taken from the standard literature on evolutionary multiobjective optimization. Results indicate that the approach is highly competitive and that can be considered a viable alternative to solve multiobjective optimization problems.
3,474 citations
TL;DR: A review of more than 90 published papers is presented here to analyze the applicability of various methods discussed and it is observed that Analytical Hierarchy Process is the most popular technique followed by outranking techniques PROMETHEE and ELECTRE.
Abstract: Multi-Criteria Decision Making (MCDM) techniques are gaining popularity in sustainable energy management. The techniques provide solutions to the problems involving conflicting and multiple objectives. Several methods based on weighted averages, priority setting, outranking, fuzzy principles and their combinations are employed for energy planning decisions. A review of more than 90 published papers is presented here to analyze the applicability of various methods discussed. A classification on application areas and the year of application is presented to highlight the trends. It is observed that Analytical Hierarchy Process is the most popular technique followed by outranking techniques PROMETHEE and ELECTRE. Validation of results with multiple methods, development of interactive decision support systems and application of fuzzy methods to tackle uncertainties in the data is observed in the published literature.
1,715 citations
TL;DR: Application of heuristic programming methods using evolutionary and genetic algorithms are described, along with application of neural networks and fuzzy rule-based systems for inferring reservoir system operating rules, to assess the state of the art in optimization of reservoir system management and operations.
Abstract: With construction of new large-scale water storage projects on the wane in the U.S. and other developed countries, attention must focus on improving the operational effectiveness and efficiency of existing reservoir systems for maximizing the beneficial uses of these projects. Optimal coordination of the many facets of reservoir systems requires the assistance of computer modeling tools to provide information for rational management and operational decisions. The purpose of this review is to assess the state-of-the-art in optimization of reservoir system management and operations and consider future directions for additional research and application. Optimization methods designed to prevail over the high-dimensional, dynamic, nonlinear, and stochastic characteristics of reservoir systems are scrutinized, as well as extensions into multiobjective optimization. Application of heuristic programming methods using evolutionary and genetic algorithms are described, along with application of neural networks and fuzzy rule-based systems for inferring reservoir system operating rules.
1,484 citations
Book•
05 Mar 2004
TL;DR: Theories of Vector Optimization: Optimality Notions, Scalarization, and Duality, and Mathematical Applications.
Abstract: Convex Analysis: Linear Spaces.- Maps on Linear Spaces.- Some Fundamental Theorems.- Theory of Vector Optimization: Optimality Notions.- Scalarization.- Existence Theorems.- Generalized Lagrange Multiplier Rule.- Duality.- Mathematical Applications: Vector Approximation.- Cooperative n Player Differential Games.- Engineering Applications: Theoretical Basics of Multiobjective Optimization.- Numerical Methods.- Multiobjective Design Problems.- Extensions to Set Optimization: Basic Concepts and Results of Set Optimization.- Contingent Epiderivatives.- Subdifferential.- Optimality Conditions.
725 citations
01 Jan 2004
TL;DR: This work states that evolutionary multiobjective optimization has become established as a separate subdiscipline combining the fields of evolutionary computation and classical multiple criteria decision making.
Abstract: Multiple, often conflicting objectives arise naturally in most real-world optimization scenarios. As evolutionary algorithms possess several characteristics that are desirable for this type of problem, this class of search strategies has been used for multiobjective optimization for more than a decade. Meanwhile evolutionary multiobjective optimization has become established as a separate subdiscipline combining the fields of evolutionary computation and classical multiple criteria decision making.
649 citations
TL;DR: A general classification of mathematical optimization problems is provided, followed by a matrix of applications that shows the areas in which these problems have been typically applied in process systems engineering.
566 citations
TL;DR: A suite of five test problems offering different patterns of such changes and different difficulties in tracking the dynamic Pareto-optimal front by a multiobjective optimization algorithm is presented.
Abstract: After demonstrating adequately the usefulness of evolutionary multiobjective optimization (EMO) algorithms in finding multiple Pareto-optimal solutions for static multiobjective optimization problems, there is now a growing need for solving dynamic multiobjective optimization problems in a similar manner. In this paper, we focus on addressing this issue by developing a number of test problems and by suggesting a baseline algorithm. Since in a dynamic multiobjective optimization problem, the resulting Pareto-optimal set is expected to change with time (or, iteration of the optimization process), a suite of five test problems offering different patterns of such changes and different difficulties in tracking the dynamic Pareto-optimal front by a multiobjective optimization algorithm is presented. Moreover, a simple example of a dynamic multiobjective optimization problem arising from a dynamic control loop is presented. An extension to a previously proposed direction-based search method is proposed for solving such problems and tested on the proposed test problems. The test problems introduced in this paper should encourage researchers interested in multiobjective optimization and dynamic optimization problems to develop more efficient algorithms in the near future.
557 citations
18 Sep 2004
TL;DR: In this article, a modified multi-objective evolutionary algorithm is introduced to focus search on these knee regions, resulting in a smaller set of solutions which are likely to be more relevant to the decision maker.
Abstract: Many real-world optimization problems have several, usually conflicting objectives. Evolutionary multi-objective optimization usually solves this predicament by searching for the whole Pareto-optimal front of solutions, and relies on a decision maker to finally select a single solution. However, in particular if the number of objectives is large, the number of Pareto-optimal solutions may be huge, and it may be very difficult to pick one “best” solution out of this large set of alternatives. As we argue in this paper, the most interesting solutions of the Pareto-optimal front are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. These solutions are sometimes also called “knees”. We then introduce a new modified multi-objective evolutionary algorithm which is able to focus search on these knee regions, resulting in a smaller set of solutions which are likely to be more relevant to the decision maker.
432 citations
01 Jan 2004
TL;DR: In this article, a modified multi-objective evolutionary algorithm is introduced to focus search on these knee regions, resulting in a smaller set of solutions which are likely to be more relevant to the decision maker.
Abstract: Many real-world optimization problems have several, usually conflicting objectives. Evolutionary multi-objective optimization usually solves this predicament by searching for the whole Pareto-optimal front of solutions, and relies on a decision maker to finally select a single solution. However, in particular if the number of objectives is large, the number of Pareto-optimal solutions may be huge, and it may be very difficult to pick one “best” solution out of this large set of alternatives. As we argue in this paper, the most interesting solutions of the Pareto-optimal front are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. These solutions are sometimes also called “knees”. We then introduce a new modified multi-objective evolutionary algorithm which is able to focus search on these knee regions, resulting in a smaller set of solutions which are likely to be more relevant to the decision maker.
413 citations
TL;DR: A two-phase fuzzy decision-making method is presented and by means of application of it to a numerical example, proved effective in providing a compromised solution in an uncertain multi-echelon supply chain network.
TL;DR: To show how applicable direct search, stochastic, and gradient-based optimization algorithms are for solving optimization problems for building design and control, this work compares the performance of these algorithms in minimizing cost functions with different smoothness.
19 Apr 2004
TL;DR: Different fuzzy-based definitions of optimality and dominated solutions, being nonpreference based, are introduced and tested on analytical test cases, in order to show their validity and nearness to human decision making.
Abstract: When dealing with many-objectives optimization problems, the concepts of Pareto-optimality and Pareto-dominance are often inefficient in modeling and simulating human decision making. This leads to an unpractical size for the set of Pareto-optimal (PO) solutions, and an additional selection criteria among solutions is usually arbitrarily considered. In the paper, different fuzzy-based definitions of optimality and dominated solutions, being nonpreference based, are introduced and tested on analytical test cases, in order to show their validity and nearness to human decision making. Based on this definitions, different subsets of PO solution set can be computed using simple and clear information provided by the decision maker and using a parameter value ranging from zero to one. When the value of the above parameter is zero, the introduced definitions coincide with classical Pareto-optimality and dominance. When the parameter value is increased, different subset of PO solutions can be obtained corresponding to higher degrees of optimality.
TL;DR: The normal constraint method is offered, which is a simple approach for generating Pareto solutions that are evenly distributed in the design space of an arbitrary number of objectives, and its critical distinction is defined, namely, the ability to generate a set of evenly distributed PareTo solutions over the complete Pare to frontier.
Abstract: Multiobjective optimization is rapidly becoming an invaluable tool in engineering design. A particular class of solutions to the multiobjective optimization problem is said to belong to the Pareto frontier. A Pareto solution, the set of which comprises the Pareto frontier, is optimal in the sense that any improvement in one design objective can only occur with the worsening of at least one other. Accordingly, the Pareto frontier plays an important role in engineering design—it characterizes the tradeoffs between conflicting design objectives. Some optimization methods can be used to automatically generate a set of Pareto solutions from which a final design is subjectively chosen by the designer. For this approach to be successful, the generated Pareto set must be truly representative of the complete optimal design space (Pareto frontier). In other words, the set must not overrepresent one region of the design space, or neglect others. Some commonly used methods comply with this requirement, whereas others do not. This paper offers a new phase in the development of the normal constraint method, which is a simple approach for generating Pareto solutions that are evenly distributed in the design space of an arbitrary number of objectives. The even distribution of the generated Pareto solutions can facilitate the process of developing an analytical expression for the Pareto frontier in n dimension. An even distribution of Pareto solutions also facilitates the task of choosing the most desirable (final) design from among the set of Pareto solutions. The normal constraint method bears some similarities to the normal boundary intersection and � -constraint methods. Importantly, the developments presented in this paper define its critical distinction, namely, the ability to generate a set of evenly distributed Pareto solutions over the complete Pareto frontier. Examples are provided that show the normal constraint method to perform favorably under the new developments when compared with the normal boundary intersection method, as well as with the original normal constraint method.
04 Dec 2004
TL;DR: It is demonstrated that the self-adaptive technique of Differential Evolution can be simply used for solving a multi-objective optimization problem where parameters are interdependent and rotational invariance is demonstrated.
Abstract: This paper demonstrates that the self-adaptive technique of Differential Evolution (DE) can be simply used for solving a multi-objective optimization problem where parameters are interdependent The real-coded crossover and mutation rates within the NSGA-II have been replaced with a simple Differential Evolution scheme, and results are reported on a rotated problem which has presented difficulties using existing Multi-objective Genetic Algorithms The Differential Evolution variant of the NSGA-II has demonstrated rotational invariance and superior performance over the NSGA-II on this problem.
TL;DR: This paper presents a comprehensive survey on techniques for simulation optimization with emphasis given on recent developments, and classify the existing techniques according to problem characteristics such as shape of the response surface, objective functions, and parameter spaces.
Abstract: For several decades, simulation has been used as a descriptive tool by the operations research community in the modeling and analysis of a wide variety of complex real systems. With recent developments in simulation optimization and advances in computing technology, it now becomes feasible to use simulation as a prescriptive tool in decision support systems. In this paper, we present a comprehensive survey on techniques for simulation optimization with emphasis given on recent developments. We classify the existing techniques according to problem characteristics such as shape of the response surface (global as compared to local optimization), objective functions (single or multiple objectives) and parameter spaces (discrete or continuous parameters). We discuss the major advantages and possible drawbacks of the different techniques. A comprehensive bibliography and future research directions are also provided in the paper.
TL;DR: This work proposes two intelligent evolutionary algorithms IEA and IMOEA using a novel intelligent gene collector (IGC) to solve single and multiobjective large parameter optimization problems, respectively.
Abstract: This work proposes two intelligent evolutionary algorithms IEA and IMOEA using a novel intelligent gene collector (IGC) to solve single and multiobjective large parameter optimization problems, respectively. IGC is the main phase in an intelligent recombination operator of IEA and IMOEA. Based on orthogonal experimental design, IGC uses a divide-and-conquer approach, which consists of adaptively dividing two individuals of parents into N pairs of gene segments, economically identifying the potentially better one of two gene segments of each pair, and systematically obtaining a potentially good approximation to the best one of all combinations using at most 2N fitness evaluations. IMOEA utilizes a novel generalized Pareto-based scale-independent fitness function for efficiently finding a set of Pareto-optimal solutions to a multiobjective optimization problem. The advantages of IEA and IMOEA are their simplicity, efficiency, and flexibility. It is shown empirically that IEA and IMOEA have high performance in solving benchmark functions comprising many parameters, as compared with some existing EAs.
TL;DR: In this paper, a thermal system design can be optimized using energy, economy and environment as separate objectives, and an evolutionary algorithm is used to find the surface of optimal solutions in the space defined by the three objective functions.
08 Sep 2004
TL;DR: The approach is an efficient and accurate way to obtain the Pareto mappings that optimize performance and power consumption and integration in an exploration framework with an event-driven trace-based simulator makes it possible to take account of important dynamic effects that have a great impact on mapping.
Abstract: In this paper we present an approach to multi-objective exploration of the mapping space of a mesh-based network-on-chip architecture. Based on evolutionary computing techniques, the approach is an efficient and accurate way to obtain the Pareto mappings that optimize performance and power consumption. Integration of the approach in an exploration framework with a kernel based on an event-driven trace-based simulator makes it possible to take account of important dynamic effects that have a great impact on mapping. Validation on both synthesized traffic and real applications (an MPEG-2 encoder/decoder system) confirms the efficiency, accuracy and scalability of the approach.
TL;DR: The general conclusions are that the weighted formula approach is to a large extent an ad-hoc approach for multi-objective optimization, whereas the lexicographic and the Pareto approach are more principled approaches, and therefore deserve more attention from the data mining community.
Abstract: This paper addresses the problem of how to evaluate the quality of a model built from the data in a multi-objective optimization scenario, where two or more quality criteria must be simultaneously optimized. A typical example is a scenario where one wants to maximize both the accuracy and the simplicity of a classification model or a candidate attribute subset in attribute selection. One reviews three very different approaches to cope with this problem, namely: (a) transforming the original multi-objective problem into a single-objective problem by using a weighted formula; (b) the lexicographical approach, where the objectives are ranked in order of priority; and (c) the Pareto approach, which consists of finding as many non-dominated solutions as possible and returning the set of non-dominated solutions to the user. One also presents a critical review of the case for and against each of these approaches. The general conclusions are that the weighted formula approach -- which is by far the most used in the data mining literature -- is to a large extent an ad-hoc approach for multi-objective optimization, whereas the lexicographic and the Pareto approach are more principled approaches, and therefore deserve more attention from the data mining community.
TL;DR: It is shown that high-order Pareto optimization holds significant potential as a tool that can be used in the balanced design of water resources systems.
Abstract: This study demonstrates the use of high-order Pareto optimization ~i.e., optimizing a system for more than two objectives! on a long-term monitoring ~LTM! application. The LTM application combines quantile kriging and the nondominated sorted genetic algorithm-II (NSGA-II) to successfully balance four objectives: ~1! minimizing sampling costs, ~2! maximizing the accuracy of interpo- lated plume maps, ~3! maximizing the relative accuracy of contaminant mass estimates, and ~4! minimizing estimation uncertainty. Optimizing the LTM application with respect to these objectives reduced the decision space of the problem from a total of 500 million designs to a set of 1,156 designs identified on the Pareto surface. Visualization of a total of eight designs aided in understanding and balancing the objectives of the application en route to a single compromise solution. This study shows that high-order Pareto optimization holds significant potential as a tool that can be used in the balanced design of water resources systems.
TL;DR: A cultural algorithm that uses domain knowledge to improve the performance of an evolutionary programming technique adopted for constrained optimization and builds a map of the feasible region to guide the search more efficiently is introduced.
Abstract: This paper introduces a cultural algorithm that uses domain knowledge to improve the performance of an evolutionary programming technique adopted for constrained optimization. The proposed approach extracts domain knowledge during the evolutionary process and builds a map of the feasible region to guide the search more efficiently. Additionally, in order to have a more efficient memory management scheme, the current implementation uses 2 n -trees to store this map of the feasible region. Results indicate that the approach is able to produce very competitive results with respect to other optimization techniques at a considerably lower computational cost.
TL;DR: This paper presents a review of approximative solution methods, that is, heuristics and metaheuristics designed for the solution of multiobjective combinatorial optimization problems (MOCO), and outlines trends in this area.
Abstract: In this paper we present a review of approximative solution methods, that is, heuristics and metaheuristics designed for the solution of multiobjective combinatorial optimization problems (MOCO). First, we discuss questions related to approximation in this context, such as performance ratios, bounds, and quality measures. We give some examples of heuristics proposed for the solution of MOCO problems. The main part of the paper covers metaheuristics and more precisely non-evolutionary methods. The pioneering methods and their derivatives are described in a unified way. We provide an algorithmic presentation of each of the methods together with examples of applications, extensions, and a bibliographic note. Finally, we outline trends in this area.
TL;DR: Smart Pareto sets as mentioned in this paper are a general approach to solving multiobjective optimization problems, which is based on generating a set of optimal solutions and then selecting the most attractive solution from this set as the final design.
Abstract: Multiobjective optimization is a powerful tool for resolving conflicting objectives in engineering design and numerous other fields. One general approach to solving multiobjective optimization problems involves generating a set of Pareto optimal solutions, followed by selecting the most attractive solution from this set as the final design. The success of this approach critically depends on the designer's ability to obtain, manage, and interpret the Pareto set—importantly, the size and distribution of the Pareto set. The potentially significant difficulties associated with comparing a significantly large number of Pareto designs can be circumvented when the Pareto set: (i) is adequately small, (ii) represents the complete Pareto frontier, (iii) emphasizes the regions of the Pareto frontier that entail significant tradeoff, and (iv) de-emphasizes the regions corresponding to little tradeoff. We call a Pareto set that possesses these four important and desirable properties a smart Pareto set. Specifically, ...
26 Jun 2004
TL;DR: An extension of the heuristic called “particle swarm optimization” (PSO) that is able to deal with multiobjective optimization problems that uses the concept of Pareto dominance to determine the flight direction of a particle.
Abstract: In this paper, we present an extension of the heuristic called “particle swarm optimization” (PSO) that is able to deal with multiobjective optimization problems. Our approach uses the concept of Pareto dominance to determine the flight direction of a particle and is based on the idea of having a set of sub-swarms instead of single particles. In each sub-swarm, a PSO algorithm is executed and, at some point, the different sub-swarms exchange information. Our proposed approach is validated using several test functions taken from the evolutionary multiobjective optimization literature. Our results indicate that the approach is highly competitive with respect to algorithms representative of the state-of-the-art in evolutionary multiobjective optimization.
TL;DR: A rigorous running time analysis of evolutionary algorithms on pseudo-Boolean multiobjective optimization problems reveals that for many problems, the simple algorithm SEMO is as efficient as this (1+1)-EA, but the improved variants FEMO and GEMO are provably better.
Abstract: This paper presents a rigorous running time analysis of evolutionary algorithms on pseudo-Boolean multiobjective optimization problems. We propose and analyze different population-based algorithms, the simple evolutionary multiobjective optimizer (SEMO), and two improved versions, fair evolutionary multiobjective optimizer (FEMO) and greedy evolutionary multiobjective optimizer (GEMO). The analysis is carried out on two biobjective model problems, leading ones trailing zeroes (LOTZ) and count ones count zeroes (COCZ), as well as on the scalable m-objective versions mLOTZ and mCOCZ. Results on the running time of the different population-based algorithms and for an alternative approach, a multistart (1+1)-EA based on the /spl epsi/-constraint method, are derived. The comparison reveals that for many problems, the simple algorithm SEMO is as efficient as this (1+1)-EA. For some problems, the improved variants FEMO and GEMO are provably better. For the analysis, we propose and apply two general tools, an upper bound technique based on a decision space partition and a randomized graph search algorithm, which facilitate the analysis considerably.
TL;DR: A parameter-less adaptive penalty scheme for genetic algorithms applied to constrained optimization problems is proposed, using feedback from the evolutionary process to define a penalty parameter for each constraint.
Abstract: A parameter-less adaptive penalty scheme for genetic algorithms applied to constrained optimization problems is proposed. Using feedback from the evolutionary process the procedure automatically defines a penalty parameter for each constraint. The user is thus relieved from the burden of having to determine sensitive parameter(s) when dealing with every new constrained optimization problem. The procedure is shown to be effective and robust when applied to test problems from the evolutionary computation literature as well as several optimization problems from the structural engineering literature. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: The particle swarm algorithm is modified to detect the pareto-optimal front, and this paper shows how this can be used to solve multiobjective optimization problems.
Abstract: Real-world optimization problems often require the minimization/maximization of more than one objective, which, in general, conflict with each other. These problems (multiobjective optimization problems, vector optimization problems) are usually treated by using weighted sums or other decision-making schemes. An alternative way is to look for the pareto-optimal front. In this paper, the particle swarm algorithm is modified to detect the pareto-optimal front.
19 Jun 2004
TL;DR: This paper addresses the neural network regularization problem from a multi-objective optimization point of view with a slightly modified version of two multi- objective optimization algorithms, the dynamic weighted aggregation (DWA) method and the elitist non-dominated sorting genetic algorithm (NSGA-II).
Abstract: Regularization is an essential technique to improve generalization of neural networks Traditionally, regularization is conducted by including an additional term in the cost function of a learning algorithm One main drawback of these regularization techniques is that a hyperparameter that determines to which extension the regularization influences the learning algorithm must be determined beforehand This paper addresses the neural network regularization problem from a multi-objective optimization point of view During the optimization, both structure and parameters of the neural network will be optimized A slightly modified version of two multi-objective optimization algorithms, the dynamic weighted aggregation (DWA) method and the elitist non-dominated sorting genetic algorithm (NSGA-II) are used and compared An evolutionary multi-objective approach to neural network regularization has a number of advantages compared to the traditional methods First, a number of models with a spectrum of model complexity can be obtained in one optimization run instead of only one single solution Second, an efficient new regularization term can be introduced, which is not applicable to gradient-based learning algorithms As a natural by-product of the multi-objective optimization approach to neural network regularization, neural network ensembles can be easily constructed using the obtained networks with different levels of model complexity Thus, the model complexity of the ensemble can be adjusted by adjusting the weight of each member network in the ensemble Simulations are carried out on a test function to illustrate the feasibility of the proposed ideas
TL;DR: In this article, a multi-objective optimization approach was proposed to determine pumping rates and well locations to prevent saltwater intrusion, while satisfying desired extraction rates in coastal aquifers.