  We develop \emph{geometric optimisation} on the manifold of Hermitian
positive definite (HPD) matrices. In particular, we consider optimising two
types of cost functions: (i) geodesically convex (g-convex); and (ii)
log-nonexpansive (LN). G-convex functions are nonconvex in the usual euclidean
sense, but convex along the manifold and thus allow global optimisation. LN
functions may fail to be even g-convex, but still remain globally optimisable
due to their special structure. We develop theoretical tools to recognise and
generate g-convex functions as well as cone theoretic fixed-point optimisation
algorithms. We illustrate our techniques by applying them to maximum-likelihood
parameter estimation for elliptically contoured distributions (a rich class
that substantially generalises the multivariate normal distribution). We
compare our fixed-point algorithms with sophisticated manifold optimisation
methods and obtain notable speedups.
