Optimisation Geometry and Its Implications for Optimisation Algorithms

Abstract

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 s..