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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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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
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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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
A Hybrid MultiObjective Evolutionary Algorithm Using an Inverse Neural Network for Aircraft Control System Design
"... Abstract – This study introduces a hybrid multiobjective evolutionary algorithm (MOEA) for the optimization of aircraft control system design. The strategy suggested here is composed mainly of two stages. The first stage consists of training an Artificial Neural Network (ANN) with objective values a ..."
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Abstract – This study introduces a hybrid multiobjective evolutionary algorithm (MOEA) for the optimization of aircraft control system design. The strategy suggested here is composed mainly of two stages. The first stage consists of training an Artificial Neural Network (ANN) with objective values as inputs and decision variables as outputs to model an approximation of the inverse of the objective function used. The second stage consists of a local improvement phase in objective space preserving objectives relationships, and a mapping process to decision variables using the trained ANN. Both the hybrid MOEA and the original MOEA were applied to an aircraft control system design application for assessment. 1
Computational steering of a multiobjective evolutionary algorithm for engineering design
 Engineering Applications of Artificial Intelligence
"... This document is the author deposited version. You are advised to consult the publisher's version if you wish to cite from it. ..."
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This document is the author deposited version. You are advised to consult the publisher's version if you wish to cite from it.
Hybrid Multiobjective Genetic Algorithm with a New Adaptive Local Search Process
"... This paper is concerned with a specific brand of evolutionary algorithms: Memetic algorithms. A new local search technique with an adaptive neighborhood setting process is introduced and assessed against a set of test functions presenting different challenges. Two performance criteria were assessed: ..."
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This paper is concerned with a specific brand of evolutionary algorithms: Memetic algorithms. A new local search technique with an adaptive neighborhood setting process is introduced and assessed against a set of test functions presenting different challenges. Two performance criteria were assessed: the convergence of the achieved results towards the true Pareto fronts and their distribution. Categories and Subject Descriptors I.2.8 [Computing Methodologies]: Artificial Intelligence – problem solving, control methods, and search – heuristics methods.
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
A New Analysis of the LebMeasure Algorithm for Calculating Hypervolume Lyndon While
"... Abstract. We present a new analysis of the LebMeasure algorithm for calculating hypervolume. We prove that although it is polynomial in the number of points, LebMeasure is exponential in the number of objectives in the worst case, not polynomial as has been claimed previously. This result has import ..."
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Abstract. We present a new analysis of the LebMeasure algorithm for calculating hypervolume. We prove that although it is polynomial in the number of points, LebMeasure is exponential in the number of objectives in the worst case, not polynomial as has been claimed previously. This result has important implications for anyone planning to use hypervolume, either as a metric to compare optimisation algorithms, or as part of a diversity mechanism in an evolutionary algorithm. Keywords: Multiobjective optimisation, evolutionary computation, performance metrics, hypervolume. 1
3 Powertrain Applications Product Development,
"... Abstract. Evolutionary multicriteria optimization has traditionally concentrated on problems comprising 2 or 3 objectives. While engineering design problems can often be conveniently formulated as multiobjective optimization problems, these often comprise a relatively large number of objectives. Suc ..."
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Abstract. Evolutionary multicriteria optimization has traditionally concentrated on problems comprising 2 or 3 objectives. While engineering design problems can often be conveniently formulated as multiobjective optimization problems, these often comprise a relatively large number of objectives. Such problems pose new challenges for algorithm design, visualisation and implementation. Each of these three topics is addressed. Progressive articulation of design preferences is demonstrated to assist in reducing the region of interest for the search and, thereby, simplified the problem. Parallel coordinates have proved a useful tool for visualising many objectives in a twodimensional graph and the computational grid and wireless Personal Digital Assistants offer technological solutions to implementation difficulties arising in complex system design. 1
A Novel Preference Articulation Operator for the Evolutionary MultiObjective Optimisation of Classifiers in Concealed Weapons Detection
"... A novel preference articulation operator for the ..."