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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
Faster SMetric Calculation by Considering Dominated Hypervolume as Klee’s Measure Problem
, 2006
"... The dominated hypervolume (or Smetric) is a commonly accepted quality measure for comparing approximations of Pareto fronts generated by multiobjective optimizers. Since optimizers exist, namely evolutionary algorithms, that use the Smetric 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 Smetric) is a commonly accepted quality measure for comparing approximations of Pareto fronts generated by multiobjective optimizers. Since optimizers exist, namely evolutionary algorithms, that use the Smetric internally several times per iteration, a faster determination of the Smetric value is of essential importance. This paper describes how to consider the Smetric 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 Smetric. 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 Smetric calculation, improving the previously known bound of O(n d−1).
On the complexity of computing the hypervolume indicator
, 2007
"... The goal of multiobjective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most reallife 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 multiobjective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most reallife 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 multiobjective optimizers providing them, unary quality measures are usually applied. Among these, the hypervolume indicator (or Smetric) 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
Multiobjective evolutionary algorithms and pattern search methods for circuit design problems
 Journal of Universal Computer Science
"... Abstract: The paper concerns the design of evolutionary algorithms and pattern search methods on two circuit design problems: the multiobjective optimization of an Operational Transconductance Amplifier and of a fifthorder leapfrog filter. The experimental results obtained show that evolutionary a ..."
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Cited by 5 (1 self)
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Abstract: The paper concerns the design of evolutionary algorithms and pattern search methods on two circuit design problems: the multiobjective optimization of an Operational Transconductance Amplifier and of a fifthorder leapfrog filter. The experimental results obtained show that evolutionary algorithms are more robust and effective in terms of the quality of the solutions and computational effort than classical methods. In particular, the observed Pareto fronts determined by evolutionary algorithms has a better spread of solutions with a larger number of nondominated solutions when compared to the classical multiobjective techniques.
Objective reduction in evolutionary multiobjective optimization: Theory and applications
 EVOLUTIONARY COMPUTATION
, 2009
"... Manyobjective problems represent a major challenge in the field of evolutionary multiobjective optimization—in terms of search efficiency, computational cost, decision making, visualization, and so on. This leads to various research questions, in particular whether certain objectives can be omitte ..."
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Cited by 5 (0 self)
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Manyobjective problems represent a major challenge in the field of evolutionary multiobjective optimization—in terms of search efficiency, computational cost, decision making, visualization, and so on. This leads to various research questions, in particular whether certain objectives can be omitted in order to overcome or at least diminish the difficulties that arise when many, that is, more than three, objective functions are involved. This study addresses this question from different perspectives. First, we investigate how adding or omitting objectives affects the problem characteristics and propose a general notion of conflict between objective sets as a theoretical foundation for objective reduction. Second, we present both exact and heuristic algorithms to systematically reduce the number of objectives, while preserving as much as possible of the dominance structure of the underlying optimization problem. Third, we demonstrate the usefulness of the proposed objective reduction method in the context of both decision making and search for a radar waveform application as well as for wellknown test functions.
Novel algorithm to calculate hypervolume indicator of Pareto approximation set
, 2007
"... Hypervolume indicator is a commonly accepted quality measure for comparing Pareto approximation set generated by multiobjective optimizers. The best known algorithm to calculate it for n points in ddimensional space has a run time of O(n d/2) with special data structures. This paper presents a rec ..."
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Hypervolume indicator is a commonly accepted quality measure for comparing Pareto approximation set generated by multiobjective optimizers. The best known algorithm to calculate it for n points in ddimensional space has a run time of O(n d/2) with special data structures. This paper presents a recursive, vertexsplitting algorithm for calculating the hypervolume indicator of a set of n noncomparable points in d > 2 dimensions. It splits out multiple child hypercuboids which can not be dominated by a splitting reference point. In special, the splitting reference point is carefully chosen to minimize the number of points in the child hypercuboids. The complexity analysis shows that the proposed algorithm achieves O( ( d 2)n) time and O(dn²) space complexity in the worst case.
Robustness in hypervolumebased multiobjective search
, 2010
"... The use of quality indicators within the search has become a popular approach in the field of evolutionary multiobjective optimization. It relies on the concept to transform the original multiobjective problem into a set problem that involves a single objective function only, namely a quality indica ..."
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Cited by 2 (0 self)
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The use of quality indicators within the search has become a popular approach in the field of evolutionary multiobjective optimization. It relies on the concept to transform the original multiobjective problem into a set problem that involves a single objective function only, namely a quality indicator, reflecting the quality of a Pareto set approximation. Especially the hypervolume indicator has gained a lot of attention in this context since it is the only set quality measure known that guarantees strict monotonicity. Accordingly, various hypervolumebased search algorithms for approximating the Pareto set have been proposed, including samplingbased methods that circumvent the problem that the hypervolume is in general hard to compute. Despite these advances, there are several open research issues in indicatorbased multiobjective search when considering realworld applications—the issue of robustness is one of them. For instance with mechanical manufacturing processes, there exist unavoidable inaccuracies that prevent a desired solution to be realized with perfect precision; therefore, a solution in terms of a concrete decision vector is not associated
A Fast Manyobjective Hypervolume Algorithm using Iterated Incremental Calculations
"... [4] ) is a popular metric for comparing the performance of multiobjective evo lutionary algorithms (MOEAs). The hypervolume of a set of solutions measures the size of the portion of objective space that is dominated by those solutions collectively. Hypervol ume captures in one scalar both the clos ..."
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[4] ) is a popular metric for comparing the performance of multiobjective evo lutionary algorithms (MOEAs). The hypervolume of a set of solutions measures the size of the portion of objective space that is dominated by those solutions collectively. Hypervol ume captures in one scalar both the closeness of the solutions to the optimal set and the spread of the solutions across objective space. Hypervolume also has nicer mathematical properties than other metrics: it was the first unary metric that detects when a set of solutions X is not worse than another set X' Three fast algorithms have been proposed for calcu lating hypervolume exactly. The Hypervolume by Slic ing Objectives algorithm (HSO) [10][12] processes the objectives in a front, rather than the points. HSO di vides the nDhypervolume to be measured into separate (n 1 )Dslices through the values in one of the objectives, then it calculates the hypervolume of each slice and sums these values to derive the total. HSO's worstcase complexity is O(mnl ) The authors are with the School of Computer Science & Software Engineering, The University of Western Australia, Western Australia 6009, Australia (email: lucas@csse.uwa.edu.au;lyndon@csse.uwa.edu.au; luigi@csse.uwa.edu.au). 9781424481262/10/$26.00 ©201 0 IEEE for reordering objectives In addition, algorithms from the computational geometry field have recently been applied to hypervolume calcula tion. Beume and Rudolph adapt the Overmars and Yap algorithm Paquete et al. [16] use a geometryinspired algorithm to calculate the maxima of a point set in 3D which has shown to be optimal by Beume et al. Another recent development is the Incremental HSO algo rithm [19] (IHSO). This is an adaptation of HSO to calculate the exclusive hypervolume contribution of a point to a front. IHSO is especially useful where hypervolume is used inline within a MOEA, either for diversity calculations [20], or for archiving purposes [21], or in selection [22], [23]. However, as we will demonstrate, IHSO can also be applied iteratively to create a new method for hypervolume metric calculations. This paper makes four principal contributions. • We describe a new algorithm IIHSO (Iterated IHSO) for calculating hypervolume exactly. IIHSO applies IHSO iteratively, starting with an empty set and adding one point at a time until the entire front has been processed. The idea of calculating hypervolume as a summation of exclusive hypervolumes was introduced by LebMea sure • We describe heuristics designed to optimise the typical performance of IIHSO, mainly for choosing a good order for adding the points to the set and a good order for processing the objectives. • We show that while HOY has by far the best worstcase