Optimisation geometry and its implications for optimisation algorithms

Optimisation geometry studies the geometry of a smooth class of optimisation problems on manifolds. A focus is placed on those classes that are fibre-wise Morse, i.e., such that in all specific problem instances, the objective function is Morse. If this condition holds, optimisation can be split into two parts: a (hard) preparation stage that computes certain lookup tables, and an (easy) optimisation stage that, given parameter values, uses the lookup tables to quickly find the global optimum for the particular problem instance. In this paper we show how the fibre-wise Morse condition can be automatically checked during the preparation stage. We also implement a version of the optimisation stage, thus providing a complete demonstration of the algorithm suggested by the theory. We discuss what goes wrong when the fibre-wise Morse condition fails and put forward some preliminary ideas on how these issues might be handled.