The ideal magnetohydrodynamic (MHD) stability limits of low aspect ratio tokamak plasmas are computed numerically for plasmas with a range of cylindrical safety factors q*, normalized plasma pressures beta , elongations kappa and central safety factors q(0). Four distinct regimes are optimized, namely: (a) low-q* plasmas with q(0)=1.1 with and without a stabilizing wall, (b) low-q* plasmas with no wall and 1.1<q(0)<2, (c) high- beta , high bootstrap fraction plasmas at moderate kappa requiring a wall and edge current drive and (d) high- beta , very high bootstrap fraction plasmas with moderate to high kappa requiring a stabilizing wall but little external current drive. A stable equilibrium is found at an aspect ratio of A=1.4 and an elongation of kappa =3.0, with 99.3% of the current provided by the plasma pressure and beta =45%. Special attention is paid to the issues of numerical convergence and the proper definition of bootstrap current fraction