Prices in electricity markets are given by the dual variables associated with the supply-demand constraint in the dispatch problem. However, in unit-commitment-based day-ahead markets, these variables are less easy to obtain. A common approach relies on resolving the dispatch problem with the commitment decisions fixed and utilizing the associated dual variables. Yet, this avenue leads to inadequate revenues to generators and necessitates an uplift payment to be made by the market operator. Recently, a convex hull pricing scheme has been proposed to reduce the impact of such payments and requires the global maximization of the associated Lagrangian dual problem, which is, in general, a piecewise-affine concave function. In this paper, we present an extreme-point-based finite-termination procedure for obtaining such a global maximizer. Unlike standard subgradient schemes where an arbitrary subgradient is used, we present a novel technique where the steepest ascent direction is constructed by solving a continuous quadratic program. The scheme initiates a move along this direction with an a priori constant steplength, with the intent of reaching the boundary of the face. A backtracking scheme allows for mitigating the impact of excessively large steps. Termination of the scheme occurs when the set of subgradients contains the zero vector. Preliminary numerical tests are seen to be promising and display the finite-termination property. Furthermore, the scheme is seen to significantly outperform standard subgradient methods.