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144
Indicatorbased selection in multiobjective search
 in Proc. 8th International Conference on Parallel Problem Solving from Nature (PPSN VIII
, 2004
"... Abstract. This paper discusses how preference information of the decision maker can in general be integrated into multiobjective search. The main idea is to first define the optimization goal in terms of a binary performance measure (indicator) and then to directly use this measure in the selection ..."
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Cited by 172 (12 self)
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Abstract. This paper discusses how preference information of the decision maker can in general be integrated into multiobjective search. The main idea is to first define the optimization goal in terms of a binary performance measure (indicator) and then to directly use this measure in the selection process. To this end, we propose a general indicatorbased evolutionary algorithm (IBEA) that can be combined with arbitrary indicators. In contrast to existing algorithms, IBEA can be adapted to the preferences of the user and moreover does not require any additional diversity preservation mechanism such as fitness sharing to be used. It is shown on several continuous and discrete benchmark problems that IBEA can substantially improve on the results generated by two popular algorithms, namely NSGAII and SPEA2, with respect to different performance measures. 1
The hypervolume indicator revisited: On the design of paretocompliant indicators via weighted integration
 In International Conference on Evolutionary MultiCriterion Optimization (EMO 2007
, 2007
"... Abstract. The design of quality measures for approximations of the Paretooptimal set is of high importance not only for the performance assessment, but also for the construction of multiobjective optimizers. Various measures have been proposed in the literature with the intention to capture differe ..."
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Cited by 59 (14 self)
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Abstract. The design of quality measures for approximations of the Paretooptimal set is of high importance not only for the performance assessment, but also for the construction of multiobjective optimizers. Various measures have been proposed in the literature with the intention to capture different preferences of the decision maker. A quality measure that possesses a highly desirable feature is the hypervolume measure: whenever one approximation completely dominates another approximation, the hypervolume of the former will be greater than the hypervolume of the latter. Unfortunately, this measure—as any measure inducing a total order on the search space—is biased, in particular towards convex, inner portions of the objective space. Thus, an open question in this context is whether it can be modified such that other preferences such as a bias towards extreme solutions can be obtained. This paper proposes a methodology for quality measure design based on the hypervolume measure and demonstrates its usefulness for three types of preferences. 1
Performance Scaling of MultiObjective Evolutionary Algorithms
, 2002
"... In real world problems, one is often faced with the problem of multiple, possibly competing, goals, which should be optimized simultaneously. These competing goals give rise to a set of compromise solutions, generally denoted as Paretooptimal. If none of the objectives have preference over the othe ..."
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Cited by 57 (5 self)
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In real world problems, one is often faced with the problem of multiple, possibly competing, goals, which should be optimized simultaneously. These competing goals give rise to a set of compromise solutions, generally denoted as Paretooptimal. If none of the objectives have preference over the other, none of these tradeoff solutions can be said to be better than any other solution in the set. Multiobjective Evolutionary Algorithms (MOEAs) can find these optimal tradeoffs in order to get a set of solutions that are optimal in an overall sense.
Running performance Metrics for evolutionary multiobjective optimization
, 2002
"... It is now well established that more than one performance metrics are necessary for evaluating a multiobjective evolutionary algorithm (MOEA). Although there exist a number of performance metrics in the MOEA literature, most of them are applied to the final nondominated set obtained by an MOEA to ..."
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Cited by 54 (3 self)
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It is now well established that more than one performance metrics are necessary for evaluating a multiobjective evolutionary algorithm (MOEA). Although there exist a number of performance metrics in the MOEA literature, most of them are applied to the final nondominated set obtained by an MOEA to evaluate its performance. In this paper, we suggest a couple of running metrics  one for measuring the convergence to a reference set and other for measuring the diversity in population members at every generation of an MOEA run. Either using a known Paretooptimal front or an agglomeration of generationwise populations, the suggested metrics reveal important insights and interesting dynamics of the working of an MOEA or help provide a comparative evaluation of two or more MOEAs.
HypE: An Algorithm for Fast HypervolumeBased ManyObjective Optimization
, 2008
"... In the field of evolutionary multicriterion optimization, the hypervolume indicator is the only single set quality measure that is known to be strictly monotonic with regard to Pareto dominance: whenever a Pareto set approximation entirely dominates another one, then also the indicator value of the ..."
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Cited by 54 (5 self)
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In the field of evolutionary multicriterion optimization, the hypervolume indicator is the only single set quality measure that is known to be strictly monotonic with regard to Pareto dominance: whenever a Pareto set approximation entirely dominates another one, then also the indicator value of the former will be better. This property is of high interest and relevance for problems involving a large number of objective functions. However, the high computational effort required for hypervolume calculation has so far prevented to fully exploit the potential of this indicator; current hypervolumebased search algorithms are limited to problems with only a few objectives. This paper addresses this issue and proposes a fast search algorithm that uses Monte Carlo simulation to approximate the exact hypervolume values. The main idea is that not the actual indicator values are important, but rather the rankings of solutions induced by the hypervolume indicator. In detail, we present HypE, a hypervolume estimation algorithm for multiobjective optimization, by which the accuracy of the estimates and the available computing resources can be traded off; thereby, not only manyobjective problems become feasible with hypervolumebased search, but also the runtime can be flexibly adapted. Moreover, we show how the same principle can be used to statistically compare the outcomes of different multiobjective optimizers with respect to the hypervolume—so far, statistical testing has been restricted to scenarios with few objectives. The experimental results indicate that HypE is highly effective for manyobjective problems in comparison to existing multiobjective evolutionary algorithms. HypE is available for download at
Using Unconstrained Elite Archives for MultiObjective Optimisation
 IEEE Transactions on Evolutionary Computation
, 2001
"... MultiObjective Evolutionary Algorithms (MOEAs) have been the subject of numer ous studies over the past 20 years. Recent work has highlighted the use of an active archive of elite, nondominated solutions to improve the optimisation speed of these algorithms. ..."
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Cited by 50 (13 self)
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MultiObjective Evolutionary Algorithms (MOEAs) have been the subject of numer ous studies over the past 20 years. Recent work has highlighted the use of an active archive of elite, nondominated solutions to improve the optimisation speed of these algorithms.
A Review of Multiobjective Test Problems and a Scalable Test Problem Toolkit
 IEEE Transactions on Evolutionary Computation
, 2006
"... Abstract—When attempting to better understand the strengths and weaknesses of an algorithm, it is important to have a strong understanding of the problem at hand. This is true for the field of multiobjective evolutionary algorithms (EAs) as it is for any other field. Many of the multiobjective test ..."
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Cited by 50 (0 self)
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Abstract—When attempting to better understand the strengths and weaknesses of an algorithm, it is important to have a strong understanding of the problem at hand. This is true for the field of multiobjective evolutionary algorithms (EAs) as it is for any other field. Many of the multiobjective test problems employed in the EA literature have not been rigorously analyzed, which makes it difficult to draw accurate conclusions about the strengths and weaknesses of the algorithms tested on them. In this paper, we systematically review and analyze many problems from the EA literature, each belonging to the important class of realvalued, unconstrained, multiobjective test problems. To support this, we first introduce a set of test problem criteria, which are in turn supported by a set of definitions. Our analysis of test problems highlights a number of areas requiring attention. Not only are many test problems poorly constructed but also the important class of nonseparable problems, particularly nonseparable multimodal problems, is poorly represented. Motivated by these findings, we present a flexible toolkit for constructing welldesigned test problems. We also present empirical results demonstrating how the toolkit can be used to test an optimizer in ways that existing test suites do not. Index Terms—Evolutionary algorithms (EAs), multiobjective evolutionary algorithms, multiobjective optimization, multiobjective test problems. I.
IEMO: An interactive evolutionary multiobjective optimization tool
 In Pattern Recognition and Machine Intelligence: First International Conference (PReMI2005
, 2005
"... Abstract. With the advent of efficient techniques for multiobjective evolutionary optimization (EMO), realworld search and optimization problems are being increasingly solved for mulitple conflicting objectives. During the past decade of research and application, most emphasis has been spent on fi ..."
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Cited by 43 (9 self)
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Abstract. With the advent of efficient techniques for multiobjective evolutionary optimization (EMO), realworld search and optimization problems are being increasingly solved for mulitple conflicting objectives. During the past decade of research and application, most emphasis has been spent on finding the complete Paretooptimal set, although EMO researchers were always aware of the importance of procedures which would help choose one particular solution from the Paretooptimal set for implementation. This is also one of the main issues on which the classical and EMO philosophies are divided on. In this paper, we address this longstanding issue and suggest an interactive EMO procedure which, for the first time, will involve a decisionmaker in the evolutionary optimization process and help choose a single solution at the end. This study is the culmination of many year’s of research on EMO and would hopefully encourage both practitioners and researchers to pay more attention in viewing the multiobjective optimization as a aggregate task of optimization and decisionmaking. 1
An adaptive scalarization method in multiobjective optimization
 SIAM Journal on Optimization
, 2009
"... Abstract. This paper presents a new method for the numerical solution of nonlinear multiobjective optimization problems with an arbitrary partial ordering in the objective space induced by a closed pointed convex cone. This algorithm is based on the wellknown scalarization approach by Pascoletti a ..."
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Cited by 20 (4 self)
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Abstract. This paper presents a new method for the numerical solution of nonlinear multiobjective optimization problems with an arbitrary partial ordering in the objective space induced by a closed pointed convex cone. This algorithm is based on the wellknown scalarization approach by Pascoletti and Serafini and adaptively controls the scalarization parameters using new sensitivity results. The computed image points give a nearly equidistant approximation of the whole Pareto surface. The effectiveness of this new method is demonstrated with various test problems and an applied problem from medicine.