  We study the classic single-item auction setting of Myerson, but under the
assumption that the buyers' values for the item are distributed over finite
supports. Using strong LP duality and polyhedral theory, we rederive various
key results regarding the revenue-maximizing auction, including the
characterization through virtual welfare maximization and the optimality of
deterministic mechanisms, as well as a novel, generic equivalence between
dominant-strategy and Bayesian incentive compatibility.
  Inspired by this, we abstract our approach to handle more general auction
settings, where the feasibility space can be given by arbitrary convex
constraints, and the objective is a convex combination of revenue and social
welfare. We characterize the optimal auctions of such systems as generalized
virtual welfare maximizers, by making use of their KKT conditions, and we
present an analogue of Myerson's payment formula for general discrete
single-parameter auction settings. Additionally, we prove that total
unimodularity of the feasibility space is a sufficient condition to guarantee
the optimality of auctions with integral allocation rules.
  Finally, we demonstrate this KKT approach by applying it to a setting where
bidders are interested in buying feasible flows on trees with capacity
constraints, and provide a combinatorial description of the (randomized, in
general) optimal auction.
