  The Barrett-Cavalcanti-Lal-Maroney (BCLM) argument stands as the most
effective means of demonstrating the reality of the quantum state. Its
advantages include being derived from very few assumptions, and a robustness to
experimental error. Finding the best way to implement the argument
experimentally is an open problem, however, and involves cleverly choosing sets
of states and measurements. I show that techniques from convex optimisation
theory can be leveraged to numerically search for these sets, which then form a
recipe for experiments that allow for the strongest statements about the
ontology of the wavefunction to be made. The optimisation approach presented is
versatile, efficient and can take account of the finite errors present in any
real experiment. I find significantly improved low-cardinality sets which are
guaranteed partially optimal for a BCLM test in low Hilbert space dimension. I
further show that mixed states can be more optimal than pure states.
