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33
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 19 (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).
Approximating the least hypervolume contributor: NPhard in general, but fast in practice
, 2008
"... ..."
Multiplicative Approximations and the Hypervolume Indicator
"... Indicatorbased algorithms have become a very popular approach to solve multiobjective 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 11 (6 self)
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Indicatorbased algorithms have become a very popular approach to solve multiobjective 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 multiobjective 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.
On SetBased 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 11 (1 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 setbased multiobjective search. It discusses how preference relations on sets can be formally defined, gives examples for selected user preferences, and proposes a general, preferenceindependent 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.
Multiobjective Optimisation Using Smetric Selection: Application to threedimensional Solution Spaces
 In CEC’2005
, 2005
"... The Smetric or hypervolume measure is a distinguished quality measure for solution sets in Pareto optimisation. Once the aim to reach a high Smetric value is appointed, it seems to be promising to directly incorporate it in the optimisation algorithm. This idea has been implemented in the SMSEMOA ..."
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Cited by 10 (4 self)
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The Smetric or hypervolume measure is a distinguished quality measure for solution sets in Pareto optimisation. Once the aim to reach a high Smetric value is appointed, it seems to be promising to directly incorporate it in the optimisation algorithm. This idea has been implemented in the SMSEMOA, an evolutionary multiobjective optimisation algorithm (EMOA) using the hypervolume measure within its selection operator. Solutions are rated according to their contribution to the dominated hypervolume of the current population. Up to now, the SMSEMOA has only been applied to functions with two objectives. The work at hand extends these studies, by surveying the behaviour of the algorithm on threeobjective problems. Additionally, a new efficient algorithm for the computation of the contributions to the dominated hypervolume in threedimensional solution spaces is presented. Different variants of selection operators are proposed. Among these, a new one is presented that rates a solution concerning the number of solutions dominating it. So, solutions in less explored regions are preferred. This rating is an efficient alternative to the Smetric criterion whenever a selection among dominated solutions has to be made. Comparative studies on standard benchmark problems show that the SMSEMOA clearly outperforms other well established EMOA. First results on a challenging realworld problem have been obtained, namely the multipoint design of an airfoil involving three objectives and nonlinear constraints. Not only a clear improvement of the baseline design, but a good coverage of the Pareto front with a small, limited number of points has been achieved.
Integrating User Preferences with Particle Swarms for Multiobjective Optimization
"... This paper proposes a method to use reference points as preferences to guide a particle swarm algorithm to search towards preferred regions of the Pareto front. A decision maker can provide several reference points, specify the extent of the spread of solutions on the Pareto front as desired, or inc ..."
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Cited by 8 (4 self)
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This paper proposes a method to use reference points as preferences to guide a particle swarm algorithm to search towards preferred regions of the Pareto front. A decision maker can provide several reference points, specify the extent of the spread of solutions on the Pareto front as desired, or include any bias between the objectives as preferences within a single execution. We incorporate the reference point method into two multiobjective particle swarm algorithms, the nondominated sorting PSO, and the maximinPSO. This paper first demonstrates the usefulness of the proposed reference point based particle swarm algorithms, then compare the two algorithms using a hypervolume metric. Both particle swarm algorithms are able to converge to the preferred regions of the Pareto front using several feasible or infeasible reference points.
On the complexity of computing the hypervolume indicator
 IEEE Trans. Evolutionary Computation
"... Abstract—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 ..."
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Cited by 8 (1 self)
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Abstract—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
Don’t be greedy when calculating hypervolume contributions
 Proceedings of the 10th International Workshop on Foundations of Genetic Algorithms (FOGA 2009
, 2009
"... contributions ..."
The maximum hypervolume set yields nearoptimal approximation
 IN PROC. 12TH ANNUAL CONFERENCE ON GENETIC AND EVOLUTIONARY COMPUTATION (GECCO ’10
, 2010
"... In order to allow a comparison of (otherwise incomparable) sets, many evolutionary multiobjective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms. The most widely used indicator is the hypervolume indicator. It measures the volume of the do ..."
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Cited by 7 (4 self)
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In order to allow a comparison of (otherwise incomparable) sets, many evolutionary multiobjective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms. The most widely used indicator is the hypervolume indicator. It measures the volume of the dominated portion of the objective space. Though the hypervolume indicator is very popular, it has not been shown that maximizing the hypervolume indicator is indeed equivalent to the overall objective of finding a good approximation of the Pareto front. To address this question, we compare the optimal approximation factor with the approximation factor achieved by sets maximizing the hypervolume indicator. We bound the optimal approximation factor of n points by 1 + Θ(1/n) for arbitrary Pareto fronts. Furthermore, we prove that the same asymptotic approximation ratio is achieved by sets of n points that maximize the hypervolume indicator. This shows that the speed of convergence of the approximation ratio achieved by maximizing the hypervolume indicator is asymptotically optimal. This implies that for large values of n, sets maximizing the hypervolume indicator quickly approach the optimal approximation ratio. Moreover, our bounds show that also for relatively small values of n, sets maximizing the hypervolume indicator achieve a nearoptimal approximation ratio.