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Approximation Quality of the Hypervolume Indicator
, 2012
"... 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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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 bounded from below by a reference point. Though the hypervolume indicator is very popular, it has not been shown that maximizing the hypervolume indicator of sets of bounded size 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 ratio with the approximation ratio achieved by twodimensional sets maximizing the hypervolume indicator. We bound the optimal multiplicative approximation ratio 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. However, there is a provable gap between the two approximation ratios which is even exponential in the ratio between the largest and the smallest value of the front. We also examine the additive approximation ratio of the hypervolume indicator in two dimensions and prove that it achieves the optimal additive approximation ratio apart from a small ratio � n/(n−2), where n is the size of the population. Hence the hypervolume indicator can be used to achieve a good additive but not a good multiplicative approximation of a Pareto front. This motivates the introduction of a “logarithmic hypervolume indicator ” which provably achieves a good multiplicative approximation ratio.
The Logarithmic Hypervolume Indicator
, 2011
"... It was recently proven that sets of points maximizing the hypervolume indicator do not give a good multiplicative approximation of the Pareto front. We introduce a new“logarithmic hypervolume indicator”and prove that it achieves a closetooptimal multiplicative approximation ratio. This is experime ..."
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Cited by 5 (3 self)
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It was recently proven that sets of points maximizing the hypervolume indicator do not give a good multiplicative approximation of the Pareto front. We introduce a new“logarithmic hypervolume indicator”and prove that it achieves a closetooptimal multiplicative approximation ratio. This is experimentally verified on several benchmark functions by comparing the approximation quality of the multiobjective covariance matrix evolution strategy (MOCMAES) with the classic hypervolume indicator and the MOCMAES with the logarithmic hypervolume indicator.
Convergence of HypervolumeBased Archiving Algorithms II: Competitiveness
"... We study the convergence behavior of (µ + λ)archiving algorithms. A (µ + λ)archiving algorithm defines how to choose in each generationµchildren fromµparents andλoffspring together. Archiving algorithms have to choose individuals online without knowing future offspring. Previous studies assumed th ..."
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Cited by 4 (1 self)
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We study the convergence behavior of (µ + λ)archiving algorithms. A (µ + λ)archiving algorithm defines how to choose in each generationµchildren fromµparents andλoffspring together. Archiving algorithms have to choose individuals online without knowing future offspring. Previous studies assumed the offspring generation to be bestcase. We assume the initial population and the offspring generation to be worstcase and use the competitive ratio to measure how much smaller hypervolumes an archiving algorithm finds due to not knowing the future in advance. We prove that all archiving algorithms which increase the hypervolume in each step (if they can) are only µcompetitive. We also present a new archiving algorithm which is (4+2/µ)competitive. This algorithm not only achieves a constant competitive ratio, but is also efficiently computable. Both properties provably do not hold for the commonly used greedy archiving algorithms, for example those used in SIBEA, SMSEMOA, or the generational MOCMAES.
E.K.: Using Diversity to Guide the Search in MultiObjective Optimization
 World Scientific
, 2004
"... The overall aim in multiobjective optimization is to aid the decisionmaking process when tackling multicriteria optimization problems. In an a posteriori approach, the strategy is to produce a set of nondominated solutions that represent a good approximation to the Pareto optimal front so that the ..."
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The overall aim in multiobjective optimization is to aid the decisionmaking process when tackling multicriteria optimization problems. In an a posteriori approach, the strategy is to produce a set of nondominated solutions that represent a good approximation to the Pareto optimal front so that the decisionmakers can select the most appropriate solution. In this paper we propose the use of diversity measures to guide the search and hence, to enhance the performance of the multiobjective search algorithm. We propose the use of diversity measures to guide the search in two different ways. First, the diversity in the objective space is used as a helper objective when evaluating candidate solutions. Secondly, the diversity in the solution space is used to choose the most promising strategy to approximate the Pareto optimal front. If the diversity is low, the emphasis is on exploration. If the diversity is high, the emphasis is on exploitation. We carry out our experiments on a twoobjective optimization problem, namely space allocation in academic institutions. This is a realworld problem in which the decisionmakers want to see a set of alternative diverse solutions in order to compare them and select the most appropriate allocation. 1.
MultiObjective MaxiMin Sorting Scheme
"... Abstract. Obtaining a well distributed nondominated Pareto front is one of the key issues in multiobjective optimization algorithms. This paper proposes a new variant for the elitist selection operator to the NSGAII algorithm, which promotes well distributed nondominated fronts. The basic idea i ..."
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Abstract. Obtaining a well distributed nondominated Pareto front is one of the key issues in multiobjective optimization algorithms. This paper proposes a new variant for the elitist selection operator to the NSGAII algorithm, which promotes well distributed nondominated fronts. The basic idea is to replace the crowding distance method by a maximin technique. The proposed technique is deployed in well known test functions and compared with the crowding distance method used in the NSGAII algorithm. This comparison is performed in terms of achieved front solutions distribution by using distance performance indices. 1
Developing Parsimonious and Efficient Algorithms for Water Resources Optimization Problems
, 2012
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Multiplicative Approximations, Optimal Hypervolume Distributions, and the Choice of the Reference Point
, 2014
"... Many optimization problems arising in applications have to consider several objective functions at the same time. Evolutionary algorithms seem to be a very natural choice for dealing with multiobjective problems as the population of such an algorithm can be used to represent the tradeoffs with res ..."
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Many optimization problems arising in applications have to consider several objective functions at the same time. Evolutionary algorithms seem to be a very natural choice for dealing with multiobjective problems as the population of such an algorithm can be used to represent the tradeoffs with respect to the given objective functions. In this paper, we contribute to the theoretical understanding of evolutionary algorithms for multiobjective problems. We consider indicatorbased algorithms whose goal is to maximize the hypervolume for a given problem by distributing µ points on the Pareto front. To gain new theoretical insights into the behavior of hypervolumebased algorithms we compare their optimization goal to the goal of achieving an optimal multiplicative approximation ratio. Our studies are carried out for different Pareto front shapes of biobjective problems. For the class of linear fronts and a class of convex fronts, we prove that maximizing the hypervolume gives the best possible approximation ratio when assuming that the extreme points have to be included in both distributions of the points on the Pareto front. Furthermore, we investigate the choice of the reference point on the approximation behavior of hypervolumebased approaches and examine Pareto fronts of different shapes by numerical calculations.
Drosophila early development by a multiobjective evolutionary
, 2009
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