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 (Sperner, 1928) and a useful visual tool for characterising and e ciently 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 nding all the local minima of an objective function with expensive function evaluations e ciently 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 rst proposed by T orn (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 (Paulavi cius and Zilinskas, 2016) as well as the PSwarm and DIRECT-L1 algorithms. Furthermore SHGO is compared with the TGO, basinhopping (BH) and di erential 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/. / Dissertation (MEng)--University of Pretoria, 2017. / Chemical Engineering / MEng / Unrestricted

In this work, connectivity graphs have been studied as models of local interactions in multi-agent robotic systems. A systematic study of the space of connectivity graphs has been done from a geometric and topological point of view. Some results on the realization of connectivity graphs in their respective configuration spaces have been given. A complexity analysis of networks, from the point of view of intrinsic structural complexity, has been given. Various topological spaces in networks, as induced from their connectivity graphs, have been recognized and put into applications, such as those concerning coverage problems in sensor networks. A framework for studying dynamic connectivity graphs has been proposed. This framework has been used for several applications that include the generation of low-complexity formations as well as collaborative beamforming in sensor networks. The theory has been verified by generating extensive simulations, with the help of software tools of computational homology and semi-definite programming. Finally, several open problems and areas of further research have been identified.