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Covariance Matrix Adaptation for Multi-objective Optimization
- Evolutionary Computation
"... The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most pow-erful evolutionary algorithms for real-valued single-objective optimization. In this pa-per, we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist C ..."
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Cited by 113 (13 self)
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The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most pow-erful evolutionary algorithms for real-valued single-objective optimization. In this pa-per, we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist CMA-ES using plus-selection and step size con-trol based on a success rule. This algorithm is compared to the standard CMA-ES. The elitist CMA-ES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in sub-optimal local minima. In the new multi-objective CMA-ES (MO-CMA-ES) a population of individuals that adapt their search strategy as in the elitist CMA-ES is maintained. These are subject to multi-objective selection. The selection is based on non-dominated sorting using either the crowding-distance or the contributing hypervolume as second sorting criterion. Both the elitist single-objective CMA-ES and the MO-CMA-ES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMA-ES. The bene-fits of the new MO-CMA-ES in comparison to the well-known NSGA-II and to NSDE, a multi-objective differential evolution algorithm, are experimentally shown.
Global Optimization Algorithms -- Theory and Application
, 2011
"... This e-book is devoted to Global Optimization algorithms, which are methods for finding solutions of high quality for an incredible wide range of problems. We introduce the basic concepts of optimization and discuss features which make optimization problems difficult and thus, should be considered ..."
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Cited by 97 (26 self)
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This e-book is devoted to Global Optimization algorithms, which are methods for finding solutions of high quality for an incredible wide range of problems. We introduce the basic concepts of optimization and discuss features which make optimization problems difficult and thus, should be considered when trying to solve them. In this book, we focus on
The hypervolume indicator revisited: On the design of pareto-compliant indicators via weighted integration
- In International Conference on Evolutionary Multi-Criterion Optimization (EMO 2007
, 2007
"... Abstract. The design of quality measures for approximations of the Pareto-optimal 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 Pareto-optimal 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 approxima-tion, the hypervolume of the former will be greater than the hypervolume of the latter. Unfortunately, this measure—as any measure inducing a to-tal order on the search space—is biased, in particular towards convex, inner portions of the objective space. Thus, an open question in this con-text 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
HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization
, 2008
"... In the field of evolutionary multi-criterion 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 multi-criterion 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 hypervolume-based 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 many-objective problems become feasible with hypervolume-based 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 many-objective problems in comparison to existing multiobjective evolutionary algorithms. HypE is available for download at
Faster S-Metric Calculation by Considering Dominated Hypervolume as Klee’s Measure Problem
, 2006
"... The dominated hypervolume (or S-metric) is a commonly accepted quality measure for comparing approximations of Pareto fronts generated by multi-objective optimizers. Since optimizers exist, namely evolutionary algorithms, that use the S-metric internally several times per iteration, a faster determi ..."
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Cited by 40 (2 self)
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The dominated hypervolume (or S-metric) is a commonly accepted quality measure for comparing approximations of Pareto fronts generated by multi-objective optimizers. Since optimizers exist, namely evolutionary algorithms, that use the S-metric internally several times per iteration, a faster determination of the S-metric value is of essential importance. This paper describes how to consider the S-metric as a special case of a more general geometrical problem called Klee’s measure problem (KMP). For KMP, an algorithm exists with run time O(n logn + n d/2 log n), for n points of d ≥ 3 dimensions. This complex algorithm is adapted to the special case of calculating the S-metric. Conceptual simplifications of the implementation are concerned that save on a factor of O(logn) and establish an upper bound of O(n logn + n d/2) for the S-metric calculation, improving the previously known bound of O(n d−1).
On Set-Based Multiobjective Optimization
, 2008
"... Assuming that evolutionary multiobjective optimization (EMO) mainly deals with set problems, one can identify three core questions in this area of research: (i) how to formalize what type of Pareto set approximation is sought, (ii) how to use this information within an algorithm to efficiently sear ..."
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Cited by 24 (4 self)
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Assuming that evolutionary multiobjective optimization (EMO) mainly deals with set problems, one can identify three core questions in this area of research: (i) how to formalize what type of Pareto set approximation is sought, (ii) how to use this information within an algorithm to efficiently search for a good Pareto set approximation, and (iii) how to compare the Pareto set approximations generated by different optimizers with respect to the formalized optimization goal. There is a vast amount of studies addressing these issues from different angles, but so far only few studies can be found that consider all questions under one roof. This paper is an attempt to summarize recent developments in the EMO field within a unifying theory of set-based multiobjective search. It discusses how preference relations on sets can be formally defined, gives examples for selected user preferences, and proposes a general, preference-independent hill climber for multiobjective optimization with theoretical convergence properties. Furthermore, it shows how to use set preference relations for statistical performance assessment and provides corresponding experimental results. The proposed methodology brings together preference articulation, algorithm design, and performance assessment under one framework and thereby opens up a new perspective on EMO.
On the complexity of computing the hypervolume indicator
, 2007
"... The goal of multi-objective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most real-life problems, only an approximation to such an optimal set can be obtained within reasonable (computing) time. To compare such approxi ..."
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Cited by 24 (2 self)
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The goal of multi-objective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most real-life problems, only an approximation to such an optimal set can be obtained within reasonable (computing) time. To compare such approximations, and thereby the performance of multi-objective optimizers providing them, unary quality measures are usually applied. Among these, the hypervolume indicator (or S-metric) is of particular relevance due to its good properties. Moreover, this indicator has been successfully integrated into stochastic optimizers, such as evolutionary algorithms, where it serves as a guidance criterion for searching the parameter space. Recent results show that computing the hypervolume indicator can be seen as solving a specialized version
Pareto-, Aggregation-, and Indicator-based Methods in Many-objective Optimization
- PROC. OF EMO 2007, VOL. 4403 OF LNCS
, 2007
"... Research within the area of Evolutionary Multi-objective Optimization (EMO) focused on two- and three-dimensional objective functions, so far. Most algorithms have been developed for and tested on this limited application area. To broaden the insight in the behavior of EMO algorithms (EMOA) in hig ..."
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Cited by 19 (1 self)
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Research within the area of Evolutionary Multi-objective Optimization (EMO) focused on two- and three-dimensional objective functions, so far. Most algorithms have been developed for and tested on this limited application area. To broaden the insight in the behavior of EMO algorithms (EMOA) in higher dimensional objective spaces, a comprehensive benchmarking is presented, featuring several state-of-the-art EMOA, as well as an aggregative approach and a restart strategy on established scalable test problems with three to six objectives. It is demonstrated why the performance of well-established EMOA (NSGA-II, SPEA2) rapidly degradates with increasing dimension. Newer EMOA like ε-MOEA, MSOPS, IBEA and SMS-EMOA cope very well with high-dimensional objective spaces. Their specific advantages and drawbacks are illustrated, thus giving valuable hints for practitioners which EMOA to choose depending on the optimization scenario. Additionally, a new method for the generation of weight vectors usable in aggregation methods is presented.
Multiplicative Approximations and the Hypervolume Indicator
"... Indicator-based algorithms have become a very popular approach to solve multi-objective optimization problems. In this paper, we contribute to the theoretical understanding of algorithms maximizing the hypervolume for a given problem by distributing µ points on the Pareto front. We examine this comm ..."
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Cited by 17 (7 self)
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Indicator-based algorithms have become a very popular approach to solve multi-objective optimization problems. In this paper, we contribute to the theoretical understanding of algorithms maximizing the hypervolume for a given problem by distributing µ points on the Pareto front. We examine this common approach with respect to the achieved multiplicative approximation ratio for a given multi-objective problem and relate it to a set of µ points on the Pareto front that achieves the best possible approximation ratio. For the class of linear fronts and a class of concave fronts, we prove that the hypervolume gives the best possible approximation ratio. In addition, we examine Pareto fronts of different shapes by numerical calculations and show that the approximation computed by the hypervolume may differ from the optimal approximation ratio.
ParadisEO-MOEO: A framework for evolutionary multi-objective optimization
, 2007
"... This paper presents ParadisEO-MOEO, a white-box objectoriented generic framework dedicated to the flexible design of evolutionary multi-objective algorithms. This paradigm-free software embeds some features and techniques for Pareto-based resolution and aims to provide a set of classes allowing to ..."
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Cited by 15 (4 self)
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This paper presents ParadisEO-MOEO, a white-box objectoriented generic framework dedicated to the flexible design of evolutionary multi-objective algorithms. This paradigm-free software embeds some features and techniques for Pareto-based resolution and aims to provide a set of classes allowing to ease and speed up the development of computationally efficient programs. It is based on a clear conceptual distinction between the solution methods and the multi-objective problems they are intended to solve. This separation confers a maximum design and code reuse. ParadisEO-MOEO provides a broad range of archive-related features (such as elitism or performance metrics) and the most common Pareto-based fitness assignment strategies (MOGA, NSGA, SPEA, IBEA and more). Furthermore, parallel and distributed models as well as hybridization mechanisms can be applied to an algorithm designed within ParadisEO-MOEO using the whole version of ParadisEO. In addition, GUIMOO, a platform-independant free software dedicated to results analysis for multi-objective problems, is briefly introduced.
