The short-term Unit Commitment (UC) problem in hydro-thermal power generation is a large-scale, Mixed-Integer NonLinear Program (MINLP), which is difficult to solve efficiently, especially for large-scale instances. It is possible to approximate the nonlinear objective function of the problem by means of piecewise-linear functions, so that UC can be approximated by a Mixed-Integer Linear Program (MILP); applying the available efficient general-purpose MILP solvers to the resulting formulations, good quality solutions can be obtained in a relatively short amount of time. We build on this approach, presenting a novel way to approximating the nonlinear objective function based on a recently developed class of \emph{valid inequalities} for the problem, called ``Perspective Cuts''. At least for many realistic instances of a general basic formulation of UC, a MILP-based heuristic obtains comparable or slightly better solutions in less time when employing the new approach rather than the standard piecewise linearizations, while being not more difficult to implement and use. Furthermore, ``dynamic'' formulations, whereby the approximation is iteratively improved, provide even better results if the approximation is appropriately controlled.