My MEng degree public defense.
“Simplicial Homology Global Optimisation: A Lipschitz Global Optimisation Algorithm”
Many, if not most, problems in science and engineering can be reformulated as optimisation problems. When little or no information of the structure of the model can be used, it can be generalized to what is known as the black-box global optimisation problem.
An example of an important problem in this class is known as the energy surface problem prominent in chemical engineering, computational material sciences and quantum mechanics. In computer science, black-box optimisation routines have also proven to be equivalent to creative problem solving routines in A.I. research.
In this research the geometric interpretation of the problem was investigated and an understanding of the problem as mutli-dimensional hypersurfaces was formed. The characterisation of an arbitrary hypersurface is accomplished through sampling (“guessing”) and deducing geometric and topological facts about the problem instance. New theorems were developed that aid in finding the best (global) solution, and other sub-optimal solutions. The total number of solutions can also be found without needing to spend resources finding all of them. The proofs behinds these theorems are built on previous results from algebraic topology and the discrete exterior calculus.
The presentation slides can be downloaded here.