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14
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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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.
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 39 (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
Hypervolumebased Multiobjective Optimization: Theoretical Foundations and Practical Implications
 THEORETICAL COMPUTER SCIENCE
, 2011
"... In recent years, indicatorbased evolutionary algorithms, allowing to implicitly incorporate user preferences into the search, have become widely used in practice to solve multiobjective optimization problems. When using this type of methods, the optimization goal changes from optimizing a set of ob ..."
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Cited by 10 (4 self)
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In recent years, indicatorbased evolutionary algorithms, allowing to implicitly incorporate user preferences into the search, have become widely used in practice to solve multiobjective optimization problems. When using this type of methods, the optimization goal changes from optimizing a set of objective functions simultaneously to the singleobjective optimization goal of finding a set of µ points that maximizes the underlying indicator. Understanding the difference between these two optimization goals is fundamental when applying indicatorbased algorithms in practice. On the one hand, a characterization of the inherent optimization goal of different indicators allows the user to choose the indicator that meets her preferences. On the other hand, knowledge about the sets of µ points with optimal indicator values—socalled optimal µdistributions—can be used in performance assessment whenever the indicator is used as a performance criterion. However, theoretical studies on indicatorbased optimization are sparse. One of the most popular indicators is the weighted hypervolume indicator. It allows to guide the search towards userdefined objective space regions and at the same time has the property of being a refinement of the Pareto dominance relation with the result that maximizing the indicator results in Paretooptimal solutions only. In previous work, we theoretically investigated the unweighted hypervolume indicator in terms of a characterization of optimal µdistributions and the influence of the hypervolume’s reference point for general biobjective optimization problems. In this
A fast incremental hypervolume algorithm
 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
, 2008
"... When hypervolume is used as part of the selection or archiving process in a multiobjective evolutionary algorithm, it is necessary to determine which solutions contribute the least hypervolume to a front. Little focus has been placed on algorithms that quickly determine these solutions and there a ..."
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Cited by 9 (0 self)
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When hypervolume is used as part of the selection or archiving process in a multiobjective evolutionary algorithm, it is necessary to determine which solutions contribute the least hypervolume to a front. Little focus has been placed on algorithms that quickly determine these solutions and there are no fast algorithms designed specifically for this purpose. We describe an algorithm, IHSO, that quickly determines a solution’s contribution. Furthermore, we describe and analyse heuristics that reorder objectives to minimize the work required for IHSO to calculate a solution’s contribution. Lastly, we describe and analyze search techniques that reduce the amount of work required for solutions other than the least contributing one. Combined, these techniques allow multiobjective evolutionary algorithms to calculate hypervolume inline in increasingly complex and large fronts in many objectives.
Heuristincs for Optimising the Calculation of Hypervolume for Multiobjective Optimisation Problems
 IEEE Congress on Evolutionary Computation (CEC
, 2005
"... Abstract The fastest known algorithm for calculating the hypervolume of a set of solutions to a multiobjective optimisation problem is the HSO algorithm (Hypervolume by Slicing Objectives). However, the performance of HSO for a given front varies a lot depending on the order in which it processes t ..."
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Cited by 8 (0 self)
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Abstract The fastest known algorithm for calculating the hypervolume of a set of solutions to a multiobjective optimisation problem is the HSO algorithm (Hypervolume by Slicing Objectives). However, the performance of HSO for a given front varies a lot depending on the order in which it processes the objectives in that front. We present and evaluate two alternative heuristics that each attempt to identify a good order for processing the objectives of a given front. We show that both heuristics make a substantial difference to the performance of HSO for randomlygenerated and benchmark data in 5–9 objectives, and that they both enable HSO to reliably avoid the worstcase performance for those fronts. The enhanced HSO will enable the use of hypervolume with larger populations in more objectives. 1
On Klee’s measure problem for grounded boxes
 In Proc. ACM Symposium on Computational Geometry (SoCG ’12
, 2012
"... A wellknown problem in computational geometry is Klee’s measure problem, which asks for the volume of a union of axisaligned boxes in dspace. In this paper, we consider Klee’s measure problem for the special case where a 2dimensional orthogonal projection of all the boxes has a common corner. We ..."
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Cited by 7 (0 self)
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A wellknown problem in computational geometry is Klee’s measure problem, which asks for the volume of a union of axisaligned boxes in dspace. In this paper, we consider Klee’s measure problem for the special case where a 2dimensional orthogonal projection of all the boxes has a common corner. We call such a set of boxes 2grounded and, more generally, a set of boxes is kgrounded if in a kdimensional orthogonal projection they share a common corner. Our main result is an O(n (d−1)/2 log 2 n) time algorithm for computing Klee’s measure for a set of n 2grounded boxes. This is an improvement of roughly O ( √ n) compared to the fastest solution of the general problem. The algorithm works for kgrounded boxes, for any k ≥ 2, and in the special case of k = d, also called the hypervolume indicator problem, the time bound can be improved further by a log n factor. The key idea of our technique is to reduce the ddimensional problem to a semidynamic weighted volume problem in dimension d − 2. The weighted volume problem requires solving a combinatorial problem of maintaining the sum of ordered products, which may be of independent interest.
Articulating User Preferences in ManyObjective Problems by Sampling the Weighted Hypervolume
"... The hypervolume indicator has become popular in recent years both for performance assessment and to guide the search of evolutionary multiobjective optimizers. Two critical research topics can be emphasized with respect to hypervolumebased search: (i) the hypervolume indicator inherently introduces ..."
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Cited by 6 (2 self)
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The hypervolume indicator has become popular in recent years both for performance assessment and to guide the search of evolutionary multiobjective optimizers. Two critical research topics can be emphasized with respect to hypervolumebased search: (i) the hypervolume indicator inherently introduces a specific preference and the question is how arbitrary user preferences can be incorporated; (ii) the exact calculation of the hypervolume indicator is expensive and efficient approaches to tackle manyobjective problems are needed. In two previous studies, we addressed both issues independently: a study proposed the weighted hypervolume indicator with which userdefined preferences can be articulated; other studies exist that propose to estimate the hypervolume indicator by MonteCarlo sampling. Here, we combine these two approaches for the first time and extend them, i.e., we present an approach of sampling the weighted hypervolume to incorporate userdefined preferences into the search for problems with many objectives. In particular, we propose weight distribution functions to stress extreme solutions and to define preferred regions of the objective space in terms of socalled preference points; sampling them allows to tackle problems with many objectives. Experiments on several test functions with up to 25 objectives show the usefulness of the approach in terms of decision making and search.
A Novel Algorithm for Nondominated Hypervolumebased Multiobjective Optimization
, 2009
"... Abstract—Hypervolume indicator is a commonly accepted quality measure to assess the set of nondominated solutions obtained by an evolutionary multiobjective optimization algorithm. Recently, an emerging trend in the design of evolutionary multiobjective optimization algorithms is to directly optimi ..."
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Abstract—Hypervolume indicator is a commonly accepted quality measure to assess the set of nondominated solutions obtained by an evolutionary multiobjective optimization algorithm. Recently, an emerging trend in the design of evolutionary multiobjective optimization algorithms is to directly optimize a quality indicator. In this paper, we propose a hypervolumebased evolutionary algorithm for multiobjective optimization. There are two main contributions of our approach, on one hand, a unique fitness assignment strategy is proposed, on the other hand, we design a slicing based method to calculate the exclusive hypervolume of each individual for environmental selection. From an extensive comparative study with three other MOEAs on a number of two and three objective test problems, it is observed that the proposed algorithm has good performance in convergence and distribution.
Speeding Up ManyObjective Optimization by Monte Carlo Approximations
, 2013
"... Many stateoftheart evolutionary vector optimization algorithms compute the contributing hypervolume for ranking candidate solutions. However, with an increasing number of objectives, calculating the volumes becomes intractable. Therefore, although hypervolumebased algorithms are often the method ..."
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Cited by 1 (1 self)
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Many stateoftheart evolutionary vector optimization algorithms compute the contributing hypervolume for ranking candidate solutions. However, with an increasing number of objectives, calculating the volumes becomes intractable. Therefore, although hypervolumebased algorithms are often the method of choice for bicriteria optimization, they are regarded as not suitable for manyobjective optimization. Recently, Monte Carlo methods have been derived and analyzed for approximating the contributing hypervolume. Turning theory into practice, we employ these results in the ranking procedure of the multiobjective covariance matrix adaptation evolution strategy (MOCMAES) as an example of a stateoftheart method for vector optimization. It is empirically shown that the approximation does not impair the quality of the obtained solutions given a budget of objective function evaluations, while considerably reducing the computation time in the case of multiple objectives. These results are obtained on common benchmark functions as well as on two design optimization tasks. Thus, employing Monte Carlo approximations makes hypervolumebased algorithms applicable to manyobjective optimization.