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The perspective reformulation has been recently proposed for constructing tight relaxations of the unit commitment (UC) problem with quadratic cost functions. In this case, it has been shown that the perspective reformulation can be cast as a second-order cone program. Conic quadratic programming is based on interior-point methods and does not benefit from the warm-start capabilities of the simplex method available in commercial mixed-integer linear programming software. It is known that the perspective formulation can be approximated with arbitrary accuracy by dynamically adding gradient-type inequalities. This study presents a higher dimensional polyhedral approximation of the perspective reformulation. The proposed approximation is not based on gradient information and can be easily implemented in a modelling language; in addition, it requires defining additional variables and constraints whose number grows moderately with increasing levels of accuracy. Extensive numerical comparisons show that the polyhedral approximation of the perspective reformulation yields a tight UC formulation, which is competitive with currently known ones.