The hypervolume indicator for multi-objective optimisation: calculation

Multi-objective problems, requiring the optimisation of two or more conflicting criteria, abound in the real world. Multi-objective optimisers produce solution sets that represent the trade-offs between problem criteria. As a result of computational and space limitations, a multi-objective optimiser is often unable to retain all generated trade-off solutions and instead must endeavour to keep the solutions that best cover the trade-off front. Indicators which map these sets into unary values that can be easily compared are valuable and are used frequently in multi-objective performance assessment, or as a part of the selection operator of a multi-objective optimiser. One indicator which incorporates many mathematical properties favourable for use in multi-objective optimisation is the hypervolume indicator. Hypervolume is the n-dimensional space that is “contained” by a set of points. It encapsulates in a single unary value a measure of the spread of the solutions along the Pareto front, as well as the closeness of the solutions to the Pareto-optimal front. However, hypervolume has one serious drawback: calculating hypervolume exactly is NPhard and exponential in the number of objectives. This thesis describes research into improving the performance of hypervolume calculation and techniques for its use. One major contribution of this thesis is the introduction of two new calculation algorithms, IIHSO and WFG, which outperform existing hypervolume calculation algorithms on several forms of test data. The best performing of the two, WFG,