A simplicial homology algorithm for Lipschitz optimisation

Abstract

Un ni iv ve er rs si it ty y o of f P Pr re et to or ri ia a SynopsisThe 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 (Sperner, 1928) 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 first proposed by Törn (1986;1990;1992). It is proven that the SHGO algorithm will always outperform TGO on function evaluations if the objective function is Lipschitz smooth. In this dissertation 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 (Paulavičius anď Zilinskas, 2016) as well as the PSwarm 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 (Jones, Oliphant, Peterson, et al., 2001-) 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/.

A machine learning approach to emulation and biophysical parameter estimation with the Community Land Model, version 5

“…We also utilize other global methods such as dual annealing (Xiang et al, 1997) and simplicial homology global optimization (SHGO) (Endres et al, 2018) to verify the results. SHGO fails to converge on an optimal solution, while dual annealing produces a similar result to differential evolution but takes several orders of magnitude more iterations and function evaluations.…”

“…The reasoning is that the MME represents a variety of structural choices and can represent this source of uncertainty outside of what a single model can achieve, assuming the MME is made up of "state-of-the-art" climate models that have been validated against historical observations . However, this approach relies on the assumption that the MME is composed of climate models which are all unique representations of the true climate, while there is evidence that state-of-theart climate models such as those participating in the Coupled Model Intercomparison Project (CMIP) share components and model development processes, which implies that they may also have common limitations (Sanderson and Knutti, 2012;Eyring et al, 2019).…”

Bifurcation of magnetorheological film–substrate elastomers subjected to biaxial pre-compression and transverse magnetic fields

“…However, it is not cautious to extend this by assumption to the case of MREs under magneto-mechanical loading. Therefore, we employ the "Simplicial Homology Global Optimization" (SHGO) algorithm (Endres et al, 2018) included in scipy for two-dimensional but also one-dimensional searches for . In both cases we opt for a sampling via Sobol sequences Sobol (1967) with 20 points by default for minimization in two dimensions ( ) and 40 points for minimization in one dimension ( 1 or 2 ).…”

Applicability of literature values for Green-Ampt parameters to account for infiltration in hydrodynamic rainfall-runoff simulations in ungauged basins

“…To carry out an automatic calibration of the Green-Ampt parameters and in some cases also initial water content and the friction within predefined ranges, optimization techniques that are implemented in the SciPy package optimization were used in this study. For the results presented in this work, the Simplicial homology global optimization (SHGO) algorithm was used [11]. Within this algorithm, the global minimum of a chosen objective function is searched by varying the different calibration parameters within predefined reasonable ranges.…”