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A simplicial homology algorithm for Lipschitz optimisation

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Abstract

The simplicial homology global optimisation (SHGO) algorithm is a general purpose global optimisation algorithm based on applications of simplicial integral homology and combinatorial topology. SHGO approximates the homology groups of a complex built on a hypersurface homeomorphic to a complex on the objective function. This provides both approximations of locally convex subdomains in the search space through Sperner’s lemma and a useful visual tool for characterising and efficiently solving higher dimensional black and grey box optimisation problems. This complex is built up using sampling points within the feasible search space as vertices. The algorithm is specialised in finding all the local minima of an objective function with expensive function evaluations efficiently which is especially suitable to applications such as energy landscape exploration. SHGO was initially developed as an improvement on the topographical global optimisation (TGO) method. It is proven that the SHGO algorithm will always outperform TGO on function evaluations if the objective function is Lipschitz smooth. In this paper SHGO is applied to non-convex problems with linear and box constraints with bounds placed on the variables. Numerical experiments on linearly constrained test problems show that SHGO gives competitive results compared to TGO and the recently developed Lc-DISIMPL algorithm as well as the PSwarm, LGO and DIRECT-L1 algorithms. Furthermore SHGO is compared with the TGO, basinhopping (BH) and differential evolution (DE) global optimisation algorithms over a large selection of black-box problems with bounds placed on the variables from the SciPy benchmarking test suite. A Python implementation of the SHGO and TGO algorithms published under a MIT license can be found from https://bitbucket.org/upiamcompthermo/shgo/.

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Notes

  1. Not necessarily total function evaluations since starting points closer to the local minima may provide better performance for a given local minimisation routines.

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Acknowledgements

The authors would also like to extend our gratitude to the anonymous reviewers and the editor whose detailed reports and suggestions helped to improve the paper.

Funding

The financial assistance of the National Research Foundation (NRF) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at, are those of the authors and are not necessarily to be attributed to the NRF. (NRF Grant Number 92781 Competitive Programme for Rated Researchers (Grant Holder WW Focke)).

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  1. Institute of Applied Materials, Department of Chemical Engineering, University of Pretoria, Pretoria, South Africa

    Stefan C. Endres & Walter W. Focke

  2. Department of Chemical Engineering, University of Pretoria, Pretoria, South Africa

    Carl Sandrock

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Endres, S.C., Sandrock, C. & Focke, W.W. A simplicial homology algorithm for Lipschitz optimisation. J Glob Optim 72, 181–217 (2018). https://doi.org/10.1007/s10898-018-0645-y

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