Lucas Bradstreet

A Fast Way of Calculating Exact Hypervolumes

We describe a new algorithm WFG for calculating hypervolume exactly. WFG is based on the recently-described observation that the exclusive hypervolume of a point p relative to a set S is equal to the difference between the inclusive hypervolume of p and the hypervolume of S with each point limited by the objective values in p. WFG applies this technique iteratively over a set to calculate its hypervolume. Experiments show that WFG is substantially faster (in five or more objectives) than all previously-described algorithms that calculate hypervolume exactly.

A Fast Incremental Hypervolume Algorithm

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 solutions contribution. Furthermore, we describe and analyse heuristics that reorder objectives to minimize the work required for IHSO to calculate a solutions 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.

Heuristics for optimizing the calculation of hypervolume for multi-objective optimization problems

The fastest known algorithm for calculating the hypervolume of a set of solutions to a multi-objective optimization 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 randomly-generated and benchmark data in 5-9 objectives, and that they both enable HSO to reliably avoid the worst-case performance for those fronts. The enhanced HSO enable the use of hypervolume with larger populations in more objectives.

Maximising Hypervolume for Selection in Multi-objective Evolutionary Algorithms

When hypervolume is used as part of the selection or archiving process in a multi-objective evolutionary algorithm, the basic requirement is to choose a subset of the solutions in a non-dominated front such that the hypervolume of the subset is maximised. We describe and evaluate two algorithms to approximate this process: a greedy algorithm that assesses and eliminates solutions individually, and a local search algorithm that assesses entire subsets. We present empirical data which suggests that a hybrid approach is needed to get the best tradeoff between good results and computational cost.

Applying the WFG algorithm to calculate incremental hypervolumes

Hypervolume is increasingly being used in-line in multi-objective evolutionary algorithms, either to promote diversity, or in an archiving mechanism, or in the selection process. The usual requirement is to determine which point in a set contributes least to the hypervolume of the set, so that that point can be discarded. We describe a new exact algorithm IWFG for performing this calculation that combines two important features from other recent algorithms: the bounding trick from WFG, that accelerates calculations by generating lots of dominated points; and the best-first queuing mechanism from IHSO, that eliminates much of the calculation for most of the points in a set. Empirical results show that IWFG is significantly faster than IHSO on much experimental data in five or more objectives.

Incrementally maximising hypervolume for selection in multi-objective evolutionary algorithms

Several multi-objective evolutionary algorithms compare the hypervolumes of different sets of points during their operation, usually for selection or archiving purposes. The basic requirement is to choose a subset of a front such that the hypervolume of that subset is maximised. We describe and evaluate three new algorithms based on incremental calculations of hypervolume using the new incremental hypervolume by slicing objectives (IHSO) algorithm: two greedy algorithms that respectively add or remove one point at a time from a front, and a local search that assesses entire subsets. Empirical evidence shows that using IHSO, the greedy algorithms are generally able to out-perform the local search and perform substantially better than previously published algorithms.

Updating exclusive hypervolume contributions cheaply

Several multi-objective evolutionary algorithms compare the hypervolumes of different sets of points during their operation, usually for selection or archiving purposes. The basic requirement is to choose a subset of a front such that the hypervolume of that subset is maximised. We describe a technique that improves the performance of hypervolume contribution based front selection schemes. This technique improves performance by allowing the update of hypervolume contributions after the addition or removal of a point, where these contributions would previously require full recalculation. Empirical evidence shows that this technique reduces runtime by up 72-99% when compared to the cost of full contribution recalculation on DTLZ and random fronts.

A fast many-objective hypervolume algorithm using iterated incremental calculations

Three fast algorithms have been proposed for calculating hypervolume exactly: the Hypervolume by Slicing Objectives algorithm (HSO) optimised with heuristics designed to improve the average case; an adaptation of the Overmars and Yap algorithm for solving the Klees measure problem; and a recent algorithm by Fonseca et al. We propose a fourth algorithm IIHSO based largely on the Incremental HSO algorithm, a version of HSO adapted to calculate the exclusive hypervolume contribution of a point to a front. We give a comprehensive analysis of IIHSO and performance comparison between three state of the art algorithms, and conclude that IIHSO outperforms the others on most important and representative data in many objectives.

Map-labelling with a multi-objective evolutionary algorithm

We present a multi-objective evolutionary algorithm approach to the map-labelling problem. Map-labelling involves placing labels for sites onto a map such that the result is easy to read and usable for navigation. However, map-users vary in their priorities and capabilities: for example, sight-impaired users need to maximise font-size, whereas other users may be willing to accept smaller labels in exchange for increased clarity of bindings of labels to sites. With a multi-objective approach, we evolve a range of labellings from which users can select according to their particular circumstances. We present results from labelling two maps, including a difficult, dense map of Newcastle County in Delaware, which clearly illustrate the advantages of the multi-objective approach.

Correction to “A Fast Incremental Hypervolume Algorithm” [Dec 08 714-723]

In the above titled paper (ibid., vol. 12, no. 6, pp. 714-723, Dec. 08), there was an error in the pseudo-code for the incremental hypervolume by slicing objectives (IHSO) that might prevent its easy implementation. The corrected pseudo-code is presented here.