A semi-continuous formulation is introduced for finding bases that minimise the error in a specific output functional of a reduced-order model. The formulation is advantageous in that it can be used with arbitrary reference data and can be easily applied to nonlinear models and functionals. A general description of the approach is given; then, explicit formulations are derived for the advection-diffusion and Burgers' equations. Numerical results are given for both linear and nonlinear functionals. These show substantial reductions in error when compared with POD modes, depending on the functional considered. The optimisation of bases for a reduced-order model using an approximated governing equation is also described, for which large increases in accuracy are obtained relative to POD modes. Copyright (C) 2015 John Wiley & Sons, Ltd.