Deconstructing the Big Valley Search Space Hypothesis
The big valley hypothesis suggests that, in combinatorial optimisation, local optima of good quality are clustered and surround the global optimum. We show here that the idea of a single valley does not always hold. Instead the big valley seems to de-construct into several valleys, also called ‘funnels’ in theoretical chemistry. We use the local optima networks model and propose an effective procedure for extracting the network data. We conduct a detailed study on four selected TSP instances of moderate size and observe that the big valley decomposes into a number of sub-valleys of different sizes and fitness distributions. Sometimes the global optimum is located in the largest valley, which suggests an easy to search landscape, but this is not generally the case. The global optimum might be located in a small valley, which offers a clear and visual explanation of the increased search difficulty in these cases. Our study opens up new possibilities for analysing and visualising combinatorial landscapes as complex networks.
KeywordsFitness landscapes Local optima networks Funnels Traveling salesman problem
Thanks are due to Darrell Whitley for relevant discussions and suggesting the paper’s title. This work was supported by the UK’s Engineering and Physical Sciences Research Council [grant number EP/J017515/1].
Data Access. All data generated during this research are openly available from the Stirling Online Repository for Research Data (http://hdl.handle.net/11667/71). All data generated during this research are openly available from the Stirling Online Repository for Research Data (http://hdl.handle.net/11667/71).
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