Orbital Geometry and Group Majorisation in Optimisation

In this paper we discuss the use of group symmetries in optimisation, in particular with respect to the structure of subdifferentials and projection operators. This allows us to study the normal cone structure of orbitopes associated with group majorisations. In particular we emphasis the utility of the "fundamental chamber" as a tool to simplify analysis. This allows us to simplify and generalise results on projections onto symmetric sets, in particular, we study projections and normal cones to sparsity constraints used in sparse signal recovery and compressed sensing using this framework.