  We test an optimised hopping parameter expansion on various Z_2 lattice
scalar field models: the Ising model, a spin-one model and lambda (phi)^4. We
do this by studying the critical indices for a variety of optimisation
criteria, in a range of dimensions and with various trial actions. We work up
to seventh order, thus going well beyond previous studies. We demonstrate how
to use numerical methods to generate the high order diagrams and their
corresponding expressions. These are then used to calculate results numerically
and, in the case of the Ising model, we obtain some analytic results. We
highlight problems with several optimisation schemes and show for the best
scheme that the critical exponents are consistent with mean field results to at
least 8 significant figures. We conclude that in its present form, such
optimised lattice expansions do not seem to be capturing the non-perturbative
infra-red physics near the critical points of scalar models.
